TeV-scale gravity in Horava-Witten theory on a compact complex
  hyperbolic threefold
TeV-scale gravity in Hořava-Witten theory on a compact
complex hyperbolic threefold
Chris Austin1
33 Collins Terrace, Maryport, Cumbria CA15 8DL, England
Abstract
The field equations and boundary conditions of Hořava-Witten theory, compactified on
a smooth compact spin quotient of CH3, where CH3 denotes the hyperbolic cousin of
3, are studied in the presence of Casimir energy density terms. If the Casimir energy
densities near one boundary result in a certain constant of integration taking a value
greater than around 105 in units of the d = 11 gravitational length, a form of thick pipe
geometry is found that realizes TeV-scale gravity by the ADD mechanism, with that
boundary becoming the inner surface of the thick pipe, where we live. Three alternative
ways in which the outer surface of the thick pipe might be stabilized consistent with
the observed value of the effective d = 4 cosmological constant are considered. In the
first alternative, the outer surface is stabilized in the classical region and the constant
of integration is fixed at around 1013 in units of the d = 11 gravitational length for
consistency with the observed cosmological constant. In the second alternative, the
four observed dimensions have reduced in size down to the d = 11 gravitational length
at the outer surface, and there are Casimir effects near the outer surface. In the third
alternative, the outer surface is stabilized in the classical region by extra fluxes of the
three-form gauge field, whose four-form field strength wraps three-cycles of the compact
six-manifold times the radial dimension of the thick pipe. Some problems related to
fitting the strong/electroweak Standard Model are considered.
1Email: chris@chrisaustin.info
http://arxiv.org/abs/0704.1476v2
Contents
1 Introduction 4
2 Thick pipe geometries 11
2.1 Hořava-Witten theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 The complex hyperbolic space CH3 . . . . . . . . . . . . . . . . . . . . 25
2.3 The field equations and boundary conditions . . . . . . . . . . . . . . . 32
2.3.1 The Christoffel symbols, Riemann tensor, and Ricci tensor . . . 34
2.3.2 The Yang-Mills coupling constants in four dimensions . . . . . . 35
2.3.3 The problem of the higher order corrections to Hořava-Witten
theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3.4 The Casimir energy density corrections to the energy-momentum
tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.3.5 The orders of perturbation theory that the terms in the Casimir
energy densities occur at . . . . . . . . . . . . . . . . . . . . . . 64
2.3.6 The expansion parameter . . . . . . . . . . . . . . . . . . . . . 66
2.3.7 Witten’s topological constraint . . . . . . . . . . . . . . . . . . 71
2.3.8 The field equations and boundary conditions for the three-form
gauge field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.3.9 The field equations and boundary conditions for the metric . . . 80
2.4 Analysis of the Einstein equations and the boundary conditions for the
metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.4.1 Beyond the proximity force approximation . . . . . . . . . . . . 90
2.4.2 The region near the inner surface of the thick pipe . . . . . . . . 93
2.4.3 The boundary conditions at the inner surface of the thick pipe . 104
2.4.4 The classical solutions in the bulk . . . . . . . . . . . . . . . . . 107
2.5 Solutions with both a and b large compared to κ2/9, at the outer surface
of the thick pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
2.5.1 Newton’s constant and the cosmological constant for solutions
with the outer surface in the classical region . . . . . . . . . . . 115
2.6 Solutions with a as small as κ2/9, at the outer surface of the thick pipe 120
2.6.1 Newton’s constant and the cosmological constant . . . . . . . . 126
2.6.2 Comparison with sub-millimetre tests of Newton’s law . . . . . 132
2.6.3 Comparison with precision solar system tests of General Relativity135
2.6.4 Further consequences of the warp factor decreasing to a small
value, at the outer surface of the thick pipe . . . . . . . . . . . . 136
2.7 Stiffening by fluxes wrapping three-cycles of the compact six-manifold
times the radial dimension . . . . . . . . . . . . . . . . . . . . . . . . . 137
2.7.1 The region near the outer surface . . . . . . . . . . . . . . . . . 142
2.7.2 Newton’s constant and the cosmological constant in the presence
of the extra fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . 152
3 Smooth compact quotients of CH
, H6, H3 and S3 160
3.1 Smooth compact arithmetic quotients of CHn and Hn . . . . . . . . . 167
3.1.1 Compactness of G/Γ for the examples of arithmetic lattices . . . 174
3.1.2 Obtaining finite index torsion-free subgroups of Γ by Selberg’s
lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
3.2 Smooth compact quotients of S3 . . . . . . . . . . . . . . . . . . . . . . 191
4 The Casimir energy densities 191
4.1 The Salam-Strathee harmonic expansion method . . . . . . . . . . . . . 196
4.2 AdS5 ×CP3 compactification of d = 11 supergravity . . . . . . . . . . 214
5 E8 vacuum gauge fields and the Standard Model 222
5.1 The lightest massive modes of the supergravity multiplet . . . . . . . . 232
5.2 An SU (9) basis for E8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
5.3 Dirac quantization condition for E8 vacuum gauge fields . . . . . . . . 241
5.4 Nonexistence of models where the Abelian Hodge - de Rham monopoles
break E8 to SU (3)× (SU (2))3 × (U (1))3 . . . . . . . . . . . . . . . . . 254
5.5 Models where the Abelian Hodge - de Rham monopoles break E8 to
SU (3)× (SU (2))2 × (U (1))4 . . . . . . . . . . . . . . . . . . . . . . . . 263
5.6 Models where the Abelian Hodge - de Rham monopoles break E8 to
SU (3)× SU (2)× (U (1))5 . . . . . . . . . . . . . . . . . . . . . . . . . 270
5.7 Generalized seesaw mechanism . . . . . . . . . . . . . . . . . . . . . . . 277
1 Introduction
The observed physical universe is a very stiff structure, approximately flat up to dis-
tances larger, by a factor of 1061, than the radius of curvature that would be expected
on the basis of the Standard Model, plus General Relativity, in 3 + 1 dimensions.
Large two-dimensional structures, such as the hull of an oil tanker, are often stiffened
by structures that extend a short distance into the third dimension. So it is natural to
wonder whether compact additional spatial dimensions, not yet observed, could play
an active role in stiffening the universe.
To study the possibility of such a mechanism, I shall consider, in this paper, the
compactification of Hořava-Witten theory [1, 2] on a smooth compact spin Kähler
manifold, that is obtained from CH3, the hyperbolic cousin of CP3, by quotienting
out the free, holomorphic action of a cocompact, torsionless, discrete subgroup of the
isometry group of CH3, which is SU(3,1). I shall look for solutions that realize TeV-
scale gravity by the ADD mechanism [3, 4, 5] in a form of thick pipe geometry [6, 7, 8],
such that the two boundaries of the Hořava-Witten universe become the inner and
outer surfaces of the thick pipe, the eleventh dimension becomes the radial direction of
the thick pipe, and the diameter of the compact six-manifold increases with increasing
distance from the inner surface of the thick pipe, where we live.
The choice of a smooth compact spin quotient of CH3, rather than a Calabi-Yau
threefold [9], as the compact six-manifold M6, means that all supersymmetries are bro-
ken by the compactification. By a fundamental theorem of Mostow, known as Mostow
rigidity [10], the geometry of M6 is now completely determined by its fundamental
group, up to an overall scale factor, so that M6 has no shape moduli. There are an
infinite number of topologically distinct smooth compact quotients of CH3, but only
a finite number with |χ (M6)| up to a given value, where χ (M6) denotes the Euler
number of M6, and only a small fraction of these are likely to be spin manifolds.
The possible values of |χ (M6)| are constrained by the fact that the squares of the
Yang-Mills coupling constants, at unification, are inversely proportional to |χ (M6)|,
and by combining an estimate of the Yang-Mills coupling constants, at unification, with
an estimate by Giudice, Rattazzi, and Wells [11] of the effective expansion parameter
for quantum gravity in eleven dimensions, the upper limit on |χ (M6)| is provisionally
estimated in subsection 2.3.6, on page 66, to be around 7×104. This upper limit might
possibly be slightly increased by an effect considered by Robinson and Wilczek [12].
This limit on the value of |χ (M6)| means that TeV-scale gravity cannot be realized
simply by choosing an extremely large value of |χ (M6)|. Instead, it is necessary that
the boundary conditions at the inner surface of the thick pipe, and the Casimir energy
density corrections to the energy-momentum tensor on and near the inner surface,
result in a certain constant of integration taking a value greater than around 105 in
units of the gravitational length in eleven dimensions.
Specifically, if y denotes the geodesic distance from the inner surface of the thick
pipe, up to an additive constant, and the d = 11 metric on M6 is b2hABdxAdxB,
where b depends only on y, and hAB is the standard metric on CH
3 introduced in
subsection 2.2, on page 25, then in the main part of the bulk, where there are no
significant source terms in the Einstein equations, we find
)1.8990
, (1)
where B is a constant of integration, that for TeV-scale gravity has to have a value
greater than around 105κ2/9, where κ is the gravitational coupling constant in eleven
dimensions.
The value of B is completely determined by the region close to the inner surface
of the thick pipe, because the only other physically significant constant of integration,
which is an overall constant multiplying the warp factor that multiplies the metric
in the four extended dimensions, does not occur in any significant terms in the field
equations or boundary conditions in this region. Thus the two boundary conditions at
the inner surface fix B and b1, the value of b at the inner surface.
A perturbative mechanism by which a large value of B
could occur is identified
in subsection 2.4.2, on page 93. In essence, the bulk power law (1) holds only for
)0.6551
, which is greater than around 103, while for 1 ∼ b1
)0.6551
, we find self-consistently that
, (2)
when the Casimir energy density corrections are taken into account, as discussed in
subsection 2.3.4, on page 57, and subsection 2.4.1, on page 90. Thus there is a quantum
region of thickness greater than around 8κ2/9 adjacent to the inner surface, in which b
increases exponentially with y.
The linear relation (2) starts to round off to a broad peak at b ∼ 103κ 29 , followed
smoothly by the classical power law (1). The only requirement for obtaining the linear
relation (2) is that a certain sign is positive rather than negative, so it seems possible
that a value of B significantly larger than κ2/9 could be found for as many as fifty
percent of the smooth compact quotients of CH3 that are spin manifolds. The actual
value of b at which the quantum relation (2) transforms into the classical relation (1),
and the corresponding value of B, will be determined by how close to the self-consistent
linear relation (2) the system is set by the boundary conditions at b1 ∼ κ2/9.
This mechanism is completely perturbative, and could be tested by one-loop calcu-
lations, for smooth compact quotients of CH3 that are spin manifolds. The numerical
coefficient in the linear relation (2) is found to be ∼ 1 if b1 is at least a factor of 2 or so
larger than the minimum value allowed by the Giudice, Rattazzi, and Wells estimate
of the expansion parameter, which is b1 ≃ 0.2κ2/9. Thus it seems likely that b1 will be
somewhere in the range from 0.4κ2/9, which corresponds to |χ (M6)| ≃ 103, to 0.8κ2/9,
which corresponds to |χ (M6)| ≃ 20.
There are inevitably significant Casimir energy density terms in the energy-momen-
tum tensor on and near the inner surface of the thick pipe, due to the Hořava-Witten
relation λ ≃ 5.8κ 23 between the d = 10 Yang-Mills coupling constant λ and κ [2], and
the fact that the d = 4 Yang-Mills coupling constants at unification are not much
smaller than 1, which implies that b1 is comparable to κ
Although the mechanism for realizing TeV-scale gravity considered in this paper is
completely perturbative, it would be desirable to be able to calculate corrections beyond
one loop, and the problem of the higher order corrections to Hořava-Witten theory is
considered in subsection 2.3.3, on page 38. The derivation of type IIA superstring
theory [13] from the Cremmer-Julia-Scherk (CJS) theory of d = 11 supergravity [14]
compactified on a small S1 [1] is reviewed, and M-theory on a smooth background is
observed to be the same as the CJS theory.
The superspace constructions of higher-derivative counterterms for the CJS theory
[15, 16] are considered, and I suggest that an obstruction might exist that prevents
the geometrical transformations in superspace [17, 18] from matching the CJS super-
symmetry variations for a general solution of the CJS field equations beyond a certain
power of θ. This would mean that with the exception of the possibly unique counter-
term constructible by the superform or ectoplasm method [19, 20, 21, 22, 16, 23], the
superspace counterterms do not result in locally supersymmetric deformations of the
CJS theory, so that since the coefficient of the unique dimension 8 counterterm [24] is
fixed by cancellation of the tangent bundle anomaly on five-branes [25, 26, 27, 28, 29],
it might be possible to calculate the predictions of the CJS theory and Hořava-Witten
theory in the framework of effective field theory, without the occurrence of undeter-
mined parameters connected with the short distance completion of the theory.
If the d = 11 metric on the four observed dimensions is a2gµνdx
µdxν , where a
depends only on y, and gµν is a metric on de Sitter space with de Sitter radius equal
to 1, then in the classical region corresponding to (1), we find
a = A
)0.7753
, (3)
where A is a constant of integration whose value is determined by the region close to
the outer surface of the thick pipe. And in the quantum region corrresponding to (2),
we find
a = A1
, (4)
where the constant A1 is determined by continuity with (3) at the transition between
the classical and quantum regions, and the exponent τ is determined by the Casimir
terms in the energy-momentum tensor for the self-consistent linear relation (2).
For B ≫ κ2/9, the existence of a solution of the boundary conditions at the outer
surface with b2 =
2a2 ≫ κ2/9, where b2 and a2 are the values of b and a at the outer
surface, is demonstrated in subsection 2.5, on page 111, and this type of solution is
found in subsection 2.5.1, on page 115, to fit the observed values of Newton’s constant
and the cosmological constant for TeV-scale gravity if τ ≃ −3 and B ∼ 1013κ2/9 ∼
10−5 metres.
This type of solution does not fully satisfy the condition for a valid reduction
to a four dimensional effective action, due to the fact that a (y) decreases from the
observed de Sitter radius of around 1026 metres at the inner surface of the thick pipe,
to around 10−5 metres at the outer surface. The fact that Newton’s law is recovered
for the gravitational force between point particles on the Planck brane [30] of the first
Randall-Sundrum model [31] suggests there is a possibility that Newton’s law might
be obtained between point particles on the inner surface of the thick pipe, but this
question is not resolved in this paper.
Solutions in which a (y) has decreased to around κ2/9 at the outer surface, and there
are Casimir effects near the outer surface, are considered in subsection 2.6, on page 120.
The three observed spatial dimensions are in this case assumed to be compactified to
a smooth compact quotient of H3, whose topology is significant for the Casimir effects
near the outer surface. There is now an additional large constant of integration, Ã,
which is the analogue of B for the quantum region near the outer surface, and by
increasing τ from around −3 towards the exponent −0.7753 in the classical relation
(3), the value of B
can be reduced from around 1013 towards a limiting value of
around 105, at a cost of rapidly increasing the value of Ã
For the case when τ = −0.7753, this type of solution is demonstrated in subsection
2.6.2, on page 132, to be consistent with the precision sub-millimetre tests of Newton’s
law [32], because most of the decrease of a (y) takes place in a very narrow region
near the outer surface, so that only a fraction ∼ 10−6 of the integral that determines
Newton’s constant comes from values of y for which a (y) is smaller than around 1018
metres.
Solutions with extra fluxes of the four-form field strength of the three-form gauge
field of d = 11 supergravity [14] wrapping three-cycles of the compact six-manifold
M6 times the radial dimension are considered in subsection 2.7, on page 137. The
outer surface is in the classical region b2 ≫ κ2/9, a2 ≫ κ2/9, and there is an additional
large constant of integration, G̃, whose square corresponds to an average value of the
energy-momentum tensor of the extra fluxes.
The value of B
can again be reduced from around 1013 towards a limiting value of
around 105, by increasing τ from around −3 towards −0.7753, at a cost now of rapidly
increasing the value of G̃
. This results in greatly increasing the value of a2, so that
a2 is around 10
22 metres for τ = −0.7753, while b2 remains ∼ B. These solutions are
therefore also consistent with the precision sub-millimetre tests of Newton’s law, for τ
in a range including −0.7753.
The value of G̃ in this type of solution does not appear to be quantized, which
suggests that cosmological models involving G̃ might resemble quintessence models
[33].
Most of the results of this paper are also valid, with minor modifications, for smooth
compact spin quotients of H6, and the construction of an infinite family of smooth
compact quotients of CH3 and H6, called arithmetic quotients, which is due to Borel
and Harish-Chandra [34], is reviewed in subsection 3.1, on page 167. Non-arithmetic
smooth compact quotients of H6 have been constructed by Gromov and Piatetski-
Shapiro [35]. Non-arithmetic smooth compact quotients of CH2 have been constructed
by Mostow [36], and non-arithmetic smooth finite-volume, but non-compact, quotients
of CH3 have been constructed by Deligne and Mostow [37], but it does not at present
seem to be known whether there exist non-arithmetic smooth compact quotients of
The compact six-manifold M6 is required to be a spin manifold, because the three-
form gauge field [38] only enters the generalized spin connection through its four-form
field strength, which is well-defined globally, so there is no possibility of defining an
analogue of a spinc structure [39] in the bulk. I do not know whether any of the
arithmetic smooth compact quotients ofCH3 orH6 are spin manifolds, but the simplest
known smooth compact quotient of H4, which is called the Davis manifold [40], is both
an arithmetic quotient and a spin manifold [41, 42]. A counting argument considered
in section 3, on page 160, suggests that for sufficiently large |χ (M6)|, non-arithmetic
smooth compact quotients of H6 will exist that are spin manifolds. The value of the
integration constant B is likely to depend on the choice of the spin structure on M6.
The value of B is also affected by the presence of topologically stabilized vacuum
Yang-Mills fields tangential to M6 on the inner surface of the thick pipe, and the
further Casimir energy density terms in the energy-momentum tensor, to which they
in turn give rise. Such vacuum Yang-Mills fields also affect the four-form field strength
of the three-form gauge field of d = 11 supergravity [38, 14], due to the boundary
condition derived by Hořava and Witten [2], and this also results in terms in the
energy-momentum tensor that are significant near the inner surface of the thick pipe,
and thus affect the value of B.
By considering certain Wilson lines formed from trees of hairpins, I demonstrate
in subsection 5.3, on page 241, that integrals over closed orientable two-dimensional
surfaces in M6, of the field strengths of Yang-Mills fields in the Cartan subalgebra of
E8, whose field strengths are proportional to Hodge - de Rham harmonic two-forms, are
restricted by a form of Dirac quantization condition to lie on a certain discrete lattice
in the Cartan subalgebra of E8, and more generally, that Abelian configurations of the
E8 Yang-Mills fields, with field strengths proportional to Hodge - de Rham harmonic
two-forms, can be topologically stabilized in magnitude, and partly also in orientation
within E8, by a form of Dirac quantization condition.
Such topologically stabilized Abelian vacuum Yang-Mills fields are restricted only
by the requirements that they break E8 to the Standard Model [43, 44] in the correct
way, as studied in subsection 5.5, on page 263, and subsection 5.6, on page 270, and
that a topological constraint derived by Witten [45] is satisfied, and that the correct
spectrum of chiral fermions, namely three Standard Model generations, plus possible
singlet neutrinos, is obtained. Witten’s topological constraint ensures that the effective
field theory, in the four extended dimensions, is free of chiral anomalies.
The first of these requirements leaves a substantial amount of flexibility in the
choice of the topologically stabilized Abelian vacuum Yang-Mills fields, and Witten’s
topological constraint also leaves a substantial amount of flexibility, unless it should
happen that the symmetric trilinear form which defines the topologically invariant cup
product H2 × H2 → H4 of M6 is either positive definite or negative definite as a
bilinear form when one of its indices takes some fixed values, thus preventing Witten’s
topological constraint from being satisfied by cancellations between contributions from
different elements of the Cartan subalgebra when the free index takes one of those fixed
values. It seems reasonable to expect that this is increasingly unlikely to occur, the
larger the second Betti number of M6 is.
Now Mostow rigidity does not imply that M6 has no Kähler shape moduli, so that
h1,1, the dimension of the Dolbeault cohomology group H1,1, is equal to 1, or that the
second Betti number of M6 is small. Rather, just as with any Kähler-Einstein metric
with a nonvanishing Ricci scalar, each Kähler modulus is equal to a fixed multiple of
the corresponding element of the first Chern class. However, by a theorem of Gro-
mov [46], all the Betti numbers of M6 are bounded by a constant times |χ (M6)|. It
seems reasonable to expect that the second Betti number of M6 will be comparable to
|χ (M6)|, and thus around 104.
If the embedding of the Standard Model in E8 is such that only a small number
of types of exotic fermion could occur, then the requirements of anomaly cancellation,
which are automatically satisfied when Witten’s topological constraint is satisfied, may
already be sufficient to prevent the occurrence of exotic chiral fermions. This happens
for the embeddings of the Standard Model in E8 studied in subsection 5.6, on page 270,
where there is only one type of exotic fermion, and the only solutions of the anomaly
cancellation constraints are an integer number of Standard Model generations. In
this case there would still be a substantial amount of flexibility in the choice of the
topologically stabilized Abelian vacuum Yang-Mills fields, when all three requirements
are satisfied.
It might also be possible to introduce partially topologically stabilized Yang-Mills
instantons in SU (2) subgroups of E8, associated with non-contractible closed four-
dimensional surfaces in M6 [47], and this might be necessary for the more complicated
types of embedding of the Standard Model in E8 studied in subsection 5.5, on page
263. However, it is not certain that this is possible, because it does not seem likely
that non-contractible closed four-dimensional surfaces in M6 will be simply connected,
and it is also unclear to what extent the orientation of such SU (2) subgroups in E8
could be topologically stabilized [48].
The introduction of topologically stabilized Abelian vacuum Yang-Mills fields of
Hosotani type [49, 50, 51], with vanishing field strength, is usually associated with
a torsion element of the fundamental group of the compact six-manifold, or in other
words, a nontrivial element a such that an = 1 for some finite integer n [9]. A smooth
compact quotient of CH3 necessarily has torsionless fundamental group, due to the
fact that CH3 is the quotient of the isometry group, SU (3, 1), by its maximal compact
subgroup, SU (3) × U (1), but examples in three dimensions suggest that it might be
possible for H1 (M6,Z) to have torsion even though the fundamental group of M6 has
no torsion, and I show in subsection 5.6, on page 270, that this would be sufficient to
enable Abelian vacuum Yang-Mills fields of Hosotani type to be topologically stabilized.
The breakings of E8 to the Standard Model considered in subsections 5.5 and 5.6
partly suppress proton decay by a mechanism related to the Aranda-Carone mechanism
[52], but I do not know whether the suppression is sufficient for consistency with current
experimental limits [53, 54, 55]. The breakings also produce natural candidates for
light sterile neutrinos [56, 57] that might be relevant if the forthcoming results of the
MiniBooNE experiment [58, 59] confirm the evidence for light sterile neutrinos from
the LSND experiment [60]. The possibility that the existence of multiple oscillation
channels involving light sterile neutrinos could improve the compatibility between the
KARMEN [61] and LSND experiments was recently demonstrated in [62].
CH3 has previously been considered in the context of M theory by Kehagias and
Russo [63]. Compact hyperbolic spaces have been considered in the context of large
extra dimensions by Kaloper, March-Russell, Starkman, and Trodden [64], and by
Tabbash [65].
2 Thick pipe geometries
I shall now briefly review Hořava-Witten theory, in Subsection 2.1, on page 17, then
summarize the relevant facts about CH3, in Subsection 2.2, on page 25. The metric
ansatz is introduced, and the field equations and boundary conditions derived, in the
presence of assumed Casimir energy densities, in Subsection 2.3, on page 32, and the
equations are studied in Subsection 2.4, on page 86.
I use units such that h̄ = c = 1. The metric signature is (−,+,+, . . . ,+). The
definitions of the Riemann and Ricci tensors are chosen to agree with the conventions
of Weinberg [66]. The Riemann tensor is defined by:
[Dµ, Dν ]Vσ = −RµνστV τ = −R τµνσ Vτ (5)
Hence:
R τµνσ = ∂µΓ
νσ − ∂νΓτµσ + ΓτµρΓρνσ − ΓτνρΓρµσ (6)
where Γτµν , the Christoffel symbol of the second kind, is defined by:
Γτµν =
gτσ (∂µgνσ + ∂νgµσ − ∂σgµν) (7)
The Ricci tensor is defined by:
Rµν = R
µτν = ∂µΓ
τν − ∂τΓτµν + ΓτµρΓρτν − ΓττρΓρµν =
∂µ∂ν ln |g| − ∂τΓτµν + ΓτµρΓρτν −
Γρµν∂ρ ln |g| (8)
where g is the determinant of the metric, gµν . These conventions are consistent with
references [2, 67, 68, 69, 70, 71] on Hořava-Witten theory, but the Riemann and Ricci
tensors, as defined here, have the opposite signs to those used in Chapters 15 and 16
of [72], and the Ricci tensor also has the opposite sign, to that defined in Chapter 18
of [43].
Laboratory and astrophysical observations, excluding the hypothesized period of
inflation, in the very early universe, are consistent with an action
Stot = SEin + Svac + SSM + SDM (9)
where
SEin = −
16πGN
−ggµνRµν (10)
is the Einstein action,
Svac = −ρvac
−g = − Λ
−g (11)
is the vacuum energy, SSM is the Standard Model matter action, and SDM is the action
for the unknown dark matter, provided that the metric, gµν , is treated classically, rather
than quantum mechanically, and all contributions to the vacuum energy, other than
ρvac, are discarded. This means, in particular, that the contributions to the vacuum
energy from the VEV of the Standard Model Higgs field, the chiral symmetry breaking
condensate and possible other condensates of QCD, and vacuum Feynman diagrams of
the Standard Model fields and the dark matter fields, in the metric gµν , are all to be
discarded.
GN is Newton’s constant, with the value [43]
GN = 6.7087× 10−39 GeV−2 (12)
so that
GN = 8.1907× 10−20 GeV−1 = 1.6160× 10−35 metres.
Variation of Stot, with respect to the metric, gives Einstein’s field equations:
Rµν −
Rgµν − Λgµν + 8πGNTµν = 0 (13)
where the energy-momentum tensor, Tµν , is defined by:
T µν =
The observed large-scale structure of the universe is consistent with a Friedmann-
Robertson-Walker metric
ds2 = −dt2 +R2 (t) g̃ij (x) dxidxj (15)
where the spatial metric g̃ij (x) is maximally symmetric, and satisfies R̃ij = −2kg̃ij ,
where k = +1 for spherical spatial sections, k = 0 for flat spatial sections, and k = −1
for hyperbolic spatial sections. The large-scale structure of Tµν is consistent with a
perfect fluid form:
Tµν = pgµν + (p+ ρ) uµuν (16)
with pressure p and energy density ρ, where uµ = (1, 0, 0, 0) is the velocity vector of
the fluid in co-moving coordinates. Einstein’s equations then lead to the Friedmann-
Lemâıtre equation
8πGN ρ
−H2 = H2
where H (t) = Ṙ
is the Hubble parameter, Ṙ = dR
, ρc =
is the critical value of
ρtot = ρ + ρvac for which k vanishes, and Λ = 8πGNρvac is the cosmological constant.
The Hubble Space Telescope Key Project [73] has given the value
H−10 = 13.6± 1.4 Gyr = (1.29± 0.13)× 1026 metres (18)
for the present value of the Hubble parameter. By combining WMAP data with other
astronomical data, Spergel et al [74] give the value
= 1.02± 0.02 (19)
However, there is no theoretical restriction on the magnitude of HR = Ṙ, so this
value is consistent with any of the three possibilities k = +1, 0, or −1, although
k = −1 is disfavoured by a standard deviation. In fact, visual inspection of the lower
two panels, of Fig. 13 of [74], does not suggest any strong preference for k = +1, as
opposed to k = −1. It seems likely that the class of models considered in the present
paper will prefer k = −1 to k = +1, due to the infinitely greater variety of the smooth
compact quotients ofH3, in comparison to the smooth compact quotients of S3, and the
correspondingly improved chances of finding a quotient whose Casimir energy densities
are such that, in combination with a suitable quotient of CH3, the observed values of
GN and Λ can be fitted. There are, however, only 18 distinct topologies with k = 0,
of which ten are compact, and the remaining eight have one or more uncompactified
dimensions, [75]. This is far too small a number of distinct topologies, for there to
be any likelihood of any of them satisfying the requirements on the Casimir energy
densities, that will make it possible to fit the observed values of GN and Λ, so I do not
expect any model, of the type studied in this paper, to have k = 0. Furthermore, most
of the flat topologies have one or more shape moduli, unlike the hyperbolic topologies,
and possibly also unlike the spherical topologies. I shall therefore, for simplicity, assume
k 6= 0.
The individual values of ρ
, and ρvac
, are not so precisely measured as their sum.
Chapter 2 of [43] quotes the values of [74]: ρ
= 0.27 ± 0.04, and ρvac
= 0.73 ± 0.04.
The baryonic and dark matter contributions to ρ
are quoted as ρb
= 0.044 ± 0.004,
and ρdm
= 0.22 ± 0.04. Chapter 19 of [43] quotes the values of Tonry et al [76]:
= 0.72± 0.05, and ρ
= 0.28± 0.05, if k = 0 is assumed. And Chapter 21 of [43]
quotes best-fit values from SNe Ia and CMB data of ρ
≈ 0.3 and ρvac
≈ 0.7. Using
the middle value ρvac
= 0.72, and the above value of H−10 , we have:
Λ = 3H20
= 0.012 Gyr−2 = 1.3× 10−52 metres−2 =
= 3.4× 10−122G−1N = 5.1× 10−84 GeV2 (20)
Hence:
ρvac = Λ/(8πGN) = 3.0× 10−47 GeV4 =
2.3× 10−3 eV
If ρ is set to zero, in the Friedmann-Lemâıtre equation (17), then for Λ > 0, the
equation has the solutions R =
(t− t0), for k = +1, R = R0e
t, for
k = 0, and R =
(t− t0), for k = −1, t > t0. All three of these solutions
satisfy Rµν = −Λgµν , and all three are in fact pieces of the maximally symmetric de
Sitter space dS4 [77]. The k = +1 solution covers the full de Sitter hyperboloid, in the
global coordinates of [77], the k = 0 solution covers the future of a single point in the
t→ −∞ “boundary” of the hyperboloid, in the planar coordinates of [77], which cover
precisely half the hyperboloid, and the k = −1 solution, for t > t0, covers the future
of an ordinary point of the hyperboloid, in the hyperbolic coordinates of [77].
known as the de Sitter radius. For the measured value of Λ, the de Sitter radius is:
= 16.0 Gyr = 1.51× 1026 metres = 0.94× 1061
GN (22)
For each of the cases k = +1, 0, and −1, we can quotient the spatial sections of the
solutions by discrete subgroups of the isometry groups of the spatial sections, that
act freely, or in other words, without fixed points, on the spatial sections, in order to
obtain locally de Sitter solutions, with Rµν = −Λgµν , and non-trivial spatial topology.
For each of the cases k = +1 and k = −1, there are an infinite number of distinct
such topologies, so it seems plausible, especially for k = −1, that there will exist
topologies for which Bose - Fermi cancellations occur in the Casimir energy densities,
for the compactifications of supergravity in eleven dimensions, and supersymmetric
Yang-Mills theory in ten dimensions, on quotients with those topologies, with just
the relative precisions I will show are needed, in order for solutions involving those
quotients, together with a suitable quotient of CH3, to fit the observed values of GN
and Λ.
The action (9) is not applicable to the hypothesized period of inflation, since accord-
ing to section 19.3.5 of [43], most current models of inflation are based on an unknown
symmetry breaking involving a new scalar field, the “inflaton”. Models of the type
considered in the present paper are expected to give very different behaviour from the
standard hot big bang model, at times earlier than the time, t, at which the hot big
bang model predicts the temperature, T , in units in which Boltzmann’s constant is
equal to 1, to be comparable to the Planck mass in eleven dimensions, in the models of
the present paper. According to [78], for temperatures higher than all particle masses,
the standard hot big bang model gives
T 2 =
where Nb denotes the number of bosonic degrees of freedom that are effectively massless
at temperature T , for example the photon contributes two units to Nb, and Nf is
the corresponding number for fermions, for example electrons and positrons together
contribute four units to Nf . If we set T = 1 TeV, and count only the observed Standard
Model particles plus the graviton, so that Nb = 26, and Nf = 90, (if the neutrinos are
assumed left-handed, so that their masses are Majorana), then
2.78× 10−3 eV
= 0.709× 10−4 metres = 2.36× 10−13 seconds (24)
which is comparable to the inverse one quarter power of the observed vacuum energy
density ρvac, equation (21). I do not yet know whether models of the type studied here
have problems with initial conditions, analogous to the horizon and flatness problems,
that led to the hypothesis of inflation [78, 79, 80, 81, 82, 83]. To answer this question
it will be necessary to study cosmological versions of these models, which will involve
partial differential equations, with the time, and the radial coordinate of the thick pipe,
as independent variables. In the present paper I shall only seek solutions such that
the metric in the four observed dimensions is locally maximally symmetric, with the
correct values of Newton’s constant and the cosmological constant. Thus the metric
in the four extended dimensions will be locally de Sitter, although I will also consider
whether or not flat and AdS solutions are possible.
The aim of this section is to determine the circumstances under which the observed
values of GN and Λ can be fitted, in a certain class of compactifications of Hořava-
Witten theory. Rather than seeking supersymmetric solutions, I shall seek solutions
in which the universe is stiff and strong, in the sense that the forces, that make it big
and flat, are much stronger, than the forces that occur in any other physical process.
In addition to fitting GN and Λ, I shall also require that the gauge coupling constants
have approximately the correct values at unification, which typically means that the
E8 fine structure constant, resulting from the compactification to 3 + 1 dimensions, is
about 1
2.1 Hořava-Witten theory
Hořava-Witten theory [1, 2] is supergravity in eleven dimensions, on a manifold with
two boundaries, or, more precisely, on the orbifold M10 × S1/Z2, where M10 is a
ten-dimensional manifold. At one-loop order in the Feynman diagram expansion, it is
necessary to introduce a supersymmetric Yang-Mills theory, with gauge group E8, on
each of the ten-dimensional boundaries, in order to cancel anomalies.
The Hořava-Witten action in the bulk is the standard Cremmer-Julia-Scherk (CJS)
action [14]. In the “upstairs” picture, working on the orbifold M10 × S1/Z2, and
omitting terms quartic in the gravitino, this is:
SCJS =
IJKDJψK −
GIJKLG
IJKLMNψN + 12ψ̄
JΓKLψM
GJKLM
gI1I2...I11CI1I2I3GI4...I7GI8...I11
where gI1I2...I11 is the tensor gI1I2...I11 = 1√−gǫ
I1I2...I11 , ǫ0 1 2 ...9 10 = 1, and GIJKL =
24∂[ICJKL]. Coordinate indices I, J,K, . . . run over all directions on M11.
The Dirac matrices ΓI are 32 × 32 real matrices satisfying {ΓI ,ΓJ} = 2gIJ . A
suitable representation of the Γa, where a, b, c, . . . are local Lorentz indices, is given,
for example, in section 2.5 of [84]. The matrices ΓI1I2...In are defined by ΓI1I2...In =
Γ[I1ΓI2 . . .Γ In], so that when the indices are all different, ΓI1I2...In = ΓI1ΓI2 . . .ΓIn .
Spinor indices are written α, β, γ, . . .. The matrices Γ0ΓI1I2...In , where the index of
Γ0 is a local Lorentz index, are symmetric for n = 1, 2, 5, 6, 9, 10, and antisymmetric
for n = 0, 3, 4, 7, 8, 11. The charge conjugation matrix is the antisymmetric matrix
C = −Γ0, where the index of Γ0 is a local Lorentz index.
We note that the right-hand spinor index of a Dirac matrix transforms under lo-
cal Lorentz transformations by matrix multiplication on the right by
where λab are the local Lorentz transformation parameters, and the left-hand spinor
index of CΓI transforms by matrix multiplication on the left by −C
1 + 1
, which is equivalent to acting on the left-hand spinor index of CΓI by
matrix multiplication on the right by
. Thus the left-hand spinor index
of CΓI is an index with the Lorentz transformation properties of the right-hand index
of a Dirac matrix, so if spinor indices with the Lorentz transformation properties of the
left-hand and right-hand spinor indices of a Dirac matrix are distinguished by writing
them as upper and lower spinor indices respectively, then C acts as a “metric”, that
lowers a spinor index. This is consistent with CΓa being an invariant tensor under
SO (10, 1) local Lorentz transformations, since
(δab + λ
δγα −
δδβ −
= (CΓa)αβ , (26)
up to terms quadratic in λab, where the identity Γ
aΓcd = Γacd + ηacΓd − ηadΓc was
used. The inverse “metric” Cαβ is defined in terms of Cαβ by CαβCβγ = δαγ . The
invariant tensor (CΓa)αβ can be written as Γaαβ , since the position of the first spinor
index distinguishes it from (Γa)
β = Γ
β. All spinors in ten or eleven dimensions will
be Majorana, which for a real representation of the Dirac matrices, means real [85].
The conjugate Majorana spinor is ψ̄α = −ψβCβα = Cαβψβ.
The manifold M11 is assumed to have the topology M10 × S1. Coordinate indices
U, V,W, . . . will run over all directions on M10, and I will use the lower-case letter
y for the coordinate in the S1 direction, and also for the coordinate index in the S1
direction. There is assumed to be an orbifold fixed point at y = y1, and another one
at y = y2 > y1. All fields are periodic in the y direction, with period 2 (y2 − y1). The
bosonic fields gUV , gyy, and CUV y are even under the reflections y → (2y1 − y) and
y → (2y2 − y), and gUy and CUVW are odd. The gravitino satisfies
ΨU (y) = ΓyΨU (2y1 − y) Ψy (y) = −ΓyΨy (2y1 − y) (27)
together with the corresponding conditions, with y1 replaced by y2.
The integral over M11, in (25), includes two copies of the physical region y1 ≤ y ≤
y2, namely the original region, and its reflection in one of the two fixed point sets. I
shall adopt the viewpoint of Hořava and Witten, that it should be possible to switch,
as convenient, between the “upstairs” viewpoint, of working on the full M11, with
these reflection symmetries imposed on the fields, and the “downstairs” viewpoint, of
working on a manifold with boundary, with the topology M10 × I1, where I1 denotes
the interval y1 ≤ y ≤ y2. For this to work, it is essential, as noted in footnote 3 of [2],
that when working on the manifold with boundary, the factor 1
, in (25), should be
replaced by 2
The conditions (27) imply that the gravitino is chiral on the ten-dimensional orb-
ifold fixed point sets, which results in a gravitational anomaly, localized on the ten-
dimensional fixed point sets. Hořava and Witten argued, in [1], that this gravitational
anomaly could be cancelled by introducing an E8 supersymmetric Yang-Mills multi-
plet, on each of the ten-dimensional fixed point sets, and they studied the required
couplings in [2]. The supersymmetric Yang-Mills action, on the orbifold fixed point set
at y1, is
SYM = −
−g tr
FUV F
χ̄ΓUDUχ
. (28)
and the action at y2 is obtained from this by the substitution M101 → M102 . The action
(28) is written in Hořava and Witten’s notation, in which “tr”, for E8, denotes 1
the trace in the adjoint representation, which they denote by “Tr”. I will also use this
notation.
Hořava and Witten do not explicitly specify the normalization of the E8 generators
they use, or, equivalently, their choice of normalization of the E8 structure constants.
This needs to be determined for the present study, because in Section 5 I shall use an
SU(9) basis for E8, rather than an SO(16) basis, and the correct normalization of the
generators, in the SU(9) basis, has to be determined. It is clear from (28) that Hořava
and Witten use hermitian E8 generators, and I shall assume that their hermitian E8
generators are given by i, or alternatively −i, times antihermitian generators, normal-
ized so that, in the SO(16) basis, the E8 Lie algebra is as given in Appendix 6.A of
[72]. Specifically, let γi, 1 ≤ i ≤ 16, be a Majorana-Weyl representation of the SO(16)
gamma matrices, so the γi are real and off block diagonal, and let σij =
[γi, γj]. Then
the σij are real, antisymmetric, and block diagonal, with two 128× 128 blocks, which
are the two irreducible spinor representations of SO(16). Choose one of the two spinor
representations, say the first, and let σ̄ij denote the restriction of σij to the corre-
sponding block. Then the generators of E8 are the 120 generators Jij of SO(16), where
Jji = −Jij , together with 128 generators Qα, whose label, α, runs over the chosen spin
representation of SO(16). The commutation relations are:
[Jij , Jkl] = Jilδjk − Jjlδik − Jikδjl + Jjkδil (29)
[Jij, Qα] = (σ̄ij)αβ Qβ (30)
[Qα, Qβ] = (σ̄ij)αβ Jij (31)
We therefore find that the matrix elements of the generators, in the adjoint represen-
tation of E8, which is also the fundamental, are given by:
Jij =
−fij,pq,rs 0
0 − (σ̄ij)βγ
0 (σ̄pq)αγ
(σ̄rs)βα 0
where
fij,pq,rs =
(δiqδpsδrj − δjqδpsδri − δipδqsδrj + δjpδqsδri − δiqδprδsj + δjqδprδsi
+δipδqrδsj − δjpδqrδsi) (34)
are the SO(16) structure constants, from (29). These generators are correctly normal-
ized so that, in doing matrix multiplications with the generators (32) and (33), the
vector index pairs (p, q), and (r, s), are to be summed over the full ranges of all the
vector indices, without restrictions, so there is no restriction, for example, to p < q.
In fact, (Jij)pq,rs = −fij,pq,rs are the correctly normalized generators of SO(16), in
the adjoint representation, with Young tableau shape (1, 1), and can be obtained, al-
ternatively, by Young tableaux methods, starting from the generators for the vector
representation of SO(16), which are:
(Jij)ef = δieδjf − δjeδif (35)
Using (32) and (33), we find that:
Tr (JijJkl) =
(δilδjk − δikδjl) = −60 (δikδjl − δilδjk) = −120δij,kl (36)
Tr (QαJij) = 0 (37)
Tr (QαQβ) = (σ̄pq)αγ (σ̄pq)γβ + (σ̄rs)δα (σ̄rs)βδ = −60δαβ − 60δβα = −120δαβ (38)
where in obtaining (36) I used that, for SO(d), we have:
fij,pq,rsfkl,rs,pq = (2d− 4) (δilδjk − δikδjl) (39)
and δij,kl =
(δikδjl − δilδjk) is the unit matrix, in the space of matrices whose rows
and columns are labelled by antisymmetrized pairs of vector indices. Hence, denoting
the 248 generators (Jij , Qα) collectively by ΛA, we have:
Tr (ΛAΛB) = −120δAB (40)
On the other hand, for the vector representation (35) of SO(16), we have:
(Jij)ef (Jkl)fe = −4δij,kl (41)
Thus the trace of the square of a generator of SO(16), in the adjoint of E8, is 30 times
the trace of the square of the corresponding generator, in the vector representation of
SO(16).
Seeking to extend (28) to a locally supersymmetric action, coupled in a locally
supersymmetric manner to the bulk supergravity multiplet, Hořava and Witten found
it necessary to modify the Bianchi identity of the four-form gauge field, so that it reads:
dGyUVWX = −3
δ (y − y1) trF (1)[UV F
WX] + δ (y − y2) trF
[UV F
where dGIJKLM = 5∂[IGJKLM ], and F
UV denotes the E8 gauge fields at y = yi. This, in
turn, implies that the three-form, CIJK , is not invariant under Yang-Mills gauge trans-
formations. It also implies, in the “upstairs” picture, that GUVWX has a discontinuity,
at y = y1, given by
GUVWX = −
ǫ (y − y1) trF (1)[UV F
WX] + . . . (43)
where ǫ (x) is 1 for x > 0, and −1 for x < 0, and . . . denotes terms that are regular
near y = y1, and thus vanish at y = y1. While in the “downstairs” picture, on the
interval y1 ≤ y ≤ y2, (43) becomes a boundary condition:
GUVWX |y=y1+ = −
[UV F
WX] (44)
Corresponding results also hold in the region of y = y2.
The non-vanishing variation of the three-form, CIJK , under Yang-Mills gauge trans-
formations, now implies that the Chern-Simons term, CGG, in the Cremmer-Julia-
Scherk action (25), has a non-vanishing variation, under Yang-Mills gauge transforma-
tions. Hořava and Witten found that this non-vanishing variation, under Yang-Mills
gauge transformations, of the Cremmer-Julia-Scherk Chern-Simons term, precisely can-
cels the one-loop quantum gauge anomaly, of the Majorana-Weyl fermions in the su-
persymmetric Yang-Mills multiplets on the orbifold fixed points, provided that
λ2 = 2π
A slightly different result was found by Conrad [86], who found λ2 = 2
32π (4πκ2)
4π (2πκ2)
3 . This difference will not have a major impact on the results of the present
paper, so I shall use the Hořava-Witten result (45), and not attempt to resolve the
issue here. The relation (45) implies that
so the Yang-Mills action is of relative order κ
Having cancelled the Yang-Mills gauge anomalies, by relating the Yang-Mills cou-
pling constant to the gravitational coupling constant as just discussed, Hořava and
Witten returned to the original purpose of introducing the Yang-Mills multiplets on
the orbifold fixed points, which was to cancel the gravitational anomalies of the graviti-
nos, on the orbifold fixed points. As explained in Section 2 (i) of [1], the “irreducible”
part of the formal twelve-form, from which the gravitino anomaly in ten dimensions
is constructed, can only be cancelled by the introduction of 248 vector multiplets on
each of the orbifold fixed point hyperplanes. This requirement is fulfilled by the E8
supersymmetric Yang-Mills multiplets. Hořava and Witten then argued that, in conse-
quence of factorization properties of the remaining terms in the full gravitational and
mixed gravitational - gauge anomalies, in ten dimensions, the remaining terms in the
gravitational and mixed anomalies can all be cancelled, provided that, in the equations
(42), (43), and (44), above, the substitutions
[UV F
WX] → trF
[UV F
WX] −
trR[UV RWX] (47)
are made uniformly, where RUV is the curvature two-form, and trR[UV RWX] must be
defined, by analogy with Section 16.1 of [72], as R Y Z[UV RWX]Y Z , and provided that, in
the quantum effective action, or in other words, the generating functional of the proper
vertices [87, 88], which is, in general, a non-local functional of the fields, a certain local
term, called the bulk Green-Schwarz term, appears in the bulk, with an appropriate
finite coefficient. The required form of the bulk Green-Schwarz term was in agreement
with the form already found from a one-loop calculation for Type IIA superstrings
[89], and from anomaly cancellation for five-branes in eleven dimensions [25, 90], and
its coefficient was studied by de Alwis [91, 92] and Conrad [86].
Hořava and Witten then completed the calculation of the action at relative order
3 , and found a problem with a term in a supersymmetry variation, proportional to
δ (0). This led to a further problem, with a term in the action at relative order κ
with a coefficient proportional to δ (0). They suggested this implies that the full theory
must have a built-in cutoff, that would replace δ (0) by a finite constant times κ−
9 , for
example, by having the gauge fields propagate in a boundary layer, of thickness about
κ2/9, rather than precisely on the orbifold fixed point hyperplanes. However Moss
has presented an improved form of Hořava-Witten theory [69, 70, 71], in which the
δ (0) terms are absent. The modifications introduced by Moss include the introduction
of a supersymmetrized Gibbons-Hawking boundary term [93, 94, 95], additional terms
bilinear in the gauginos in (42), (43), and (44) above, and a modification to the chirality
conditions (27) on the gravitino, in the neighbourhood of an orbifold fixed point, which
for the components ψU , in the upstairs picture, amounts to introducing a step function
term in the behaviour of (1− Γy)ψU , near the fixed point, analogous to (43) above.
The existence of Moss’s improved form of the theory suggests it is reasonable to
assume that the Yang-Mills multiplets do not, after all, spread into a boundary layer
of nonzero thickness in the bulk, and do, indeed, stay in the orbifold fixed point hyper-
planes, of zero thickness. The study of the boundary conditions, and of the field equa-
tions in the bulk, near the boundaries, in the present paper, depend on this assumption
for their validity, so the conclusions about the existence of thick pipe geometries, and
the possibility of fitting both Newton’s constant and the cosmological constant, for
topologies such that the Casimir energy densities cancel to the required relative preci-
sions, depend on the existence of Moss’s improved form of the theory. However, these
studies do not involve the fermi fields, so I will not need to use the explicit form of the
modifications introduced by Moss.
The assumption that the Yang-Mills multiplets do, indeed, stay in the orbifold
fixed-point hyperplanes, of zero thickness, means that, for the further development of
Hořava-Witten theory, it is essential to treat the step functions, such as in (43) above,
and their derivatives, by a consistent limiting procedure, from properly regularized
versions, as discussed by Bilal and Metzger [96, 28]. However this is not necessary in
the present paper.
Lukas, Ovrut, and Waldram, [67], have pointed out that, corresponding to the
replacement (47), supersymmetry is likely to require that, in the Yang-Mills action
(28), the corresponding replacement
(i)UV → trF (i)UV F (i)UV −
RUVWXR
UVWX − 4RUVRUV +R2
= trF
(i)UV − 3R [UV[UV R
WX] (48)
is made, where RUV now denotes the Ricci tensor. This would be analogous to the
situation for the E8 × E8 heterotic superstring [97, 98], whose effective low-energy
field theory action contains the expression to the right of the arrow in (48), summed
over both the E8 groups. In this case, the Lovelock-Gauss-Bonnet term [99, 100, 101]
[UV R
WX] is stated, in Section 16.1 of [72], to be related, by supersymme-
try, to the Lorentz Chern-Simons term that is included in the field strength of the
d = 10, N = 1 supergravity two-form, by the original Green-Schwarz anomaly can-
cellation mechanism [102]. The Lovelock-Gauss-Bonnet term, for the E8 × E8 het-
erotic superstring, was found by Gross and Witten [103], by means of a low energy
expansion of tree-level superstring scattering amplitudes. The relative coefficients of
RUVWXR
UVWX, RUVR
UV , and R2 were fixed to the Lovelock-Gauss-Bonnet form by
Zwiebach [104], who pointed out this linear combination contains no terms quadratic
in the graviton, and thus does not lead to the occurrence of ghosts, in the free graviton
propagator. An analogue of the positive energy theorem [105, 106] for the Einstein
action, together with the Lovelock-Gauss-Bonnet term, as it occurs in the effective
low-energy field theory action for the E8 × E8 heterotic superstring, was proved by
Kowalski-Glikman [107], and the Lovelock-Gauss-Bonnet term was found by Candelas,
Horowitz, Strominger, and Witten [9], to make it possible to circumvent the no-go
theorem [108], for compactifications of supersymmetric Yang-Mills theory coupled to
N = 1 supergravity in ten dimensions [109, 110].
To the best of my knowledge, the corresponding Lovelock-Gauss-Bonnet terms for
Hořava-Witten theory, given by making the substitutions (48) in the Yang-Mills ac-
tion (28), have not yet been directly derived, nor explicitly related by supersymmetry
to the modified Bianchi identity (42), with the substitutions (47). This would pre-
sumably require the systematic study of Slavnov-Taylor identities [111, 112, 113] for
BRST quantized [114, 115] Hořava-Witten theory, perhaps in the Batalin-Vilkovisky
framework [116, 117, 118, 119, 120]. The Lovelock-Gauss-Bonnet terms would, then,
presumably be found as local terms, in the generating functional of proper vertices, on
the orbifold fixed-point hyperplanes, with the expected finite coefficients, in a similar
manner to the bulk Green-Schwarz term. I shall simply follow Lukas, Ovrut, and Wal-
dram [67], and assume these terms to be present, with the coefficients implied by the
substitutions (48).
2.2 The complex hyperbolic space CH
I shall assume that six of the nine spatial dimensions, of M10, are compactified on
a smooth compact spin quotient of CH3, the complex hyperbolic space with three
complex dimensions. A detailed account of the geometry of complex hyperbolic space
has been given by Goldman [121], but for the present study of the field equations and
boundary conditions, I shall only need the very simplest properties of CH3, which I
shall now summarize.
The study of CH3 is facilitated by the use of complex coordinates. I shall consider
the transformation from Cartesian coordinates to complex coordinates to be a special
case of a general coordinate transformation, and use the corresponding notation. For
2n real dimensions, we define a 2n by 2n complex matrix Uµ ν by:
U r2s−1 =
δrs, U
2s−1 =
δrs, U
δrs, U
2s = − i√2 δ
s (49)
for 1 ≤ r ≤ n and 1 ≤ s ≤ n. Then we define complex coordinates zµ by a complex
general linear transformation:
zµ = Uµ νx
ν (50)
zr = U r νx
ν = U r2s−1x
2s−1 + U r2sx
x2r−1 + ix2r
zr̄ = U r̄ νx
ν = U r̄2s−1x
2s−1 + U r̄2sx
x2r−1 − ix2r
for 1 ≤ r ≤ n. Thus zr̄ = (zr)∗, where ∗ denotes complex conjugation. We define the
inverse, V , of U , by:
V 2r−1 s =
δr s, V
δr s, V
s = − i√2δ
δr s (53)
xµ = V µ νz
ν (54)
In general, on the change to complex coordinates, a contravariant index is transformed
by U , thus xµ → Uµ νxν , and a covariant index is transformed by V , thus ∂µ →
V ν µ∂ν . The metric in flat Cartesian coordinates, namely the Kronecker delta, δµν , is
not preserved by the transformation to complex coordinates. Its components in the
complex coordinate basis, which I will denote by δ̃µν , are given by:
δ̃µν = V
νδστ (55)
Explicitly:
δ̃rs = δ̃r̄s̄ = 0, δ̃rs̄ = δ̃r̄s = δrs (56)
Thus δ̃µνz
µzν = 2zrzr̄ = δστx
σxτ , where I have introduced a summation convention
specific to complex coordinates, namely that if a holomorpic contravariant index, i.e.
an unbarred contravariant index that runs from 1 to n, has the same letter as an
antiholomorphic contravariant index, i.e. a barred contravariant index that runs from
1 to n, then the formula is to be summed over all values of that letter, from 1 to n. I
shall also use the corresponding convention when a holomorphic covariant index, and
an antiholomorphic covariant index, have the same letter, and summation from 1 to
n also applies, when a holomorphic contravariant index, and a holomorphic covariant
index, have the same letter, and it also applies, when an antiholomorphic contravariant
index, and an antiholomorphic covariant index, have the same letter.
Similarly:
δ̃µν = Uµ σU
στ (57)
The components are:
δ̃rs = δ̃r̄s̄ = 0, δ̃rs̄ = δ̃r̄s = δ
rs (58)
We then find, for example, that δ̃rµδ̃µs = δ̃
r̄µδ̃µs̄ = δ
s, δ̃
rµδ̃µs̄ = δ̃
r̄µδ̃µs = 0.
It is convenient also to define δrs̄, δr̄s, δ
rs̄, and δr̄s, (without tildes), by:
δrs̄ = δr̄s = δ
rs̄ = δr̄s = δrs = δ
rs (59)
I shall adopt the convention that, when using complex coordinates, the indices of the
coordinates are lowered and raised by the flat space complex metric, δ̃µν and δ̃
µν , not
by whatever curved metric is under consideration. We thus have:
zr = z
r̄ = (zr)
, zr̄ = z
r = (zr̄)
zrzr̄ = zrz
r = zr̄zr̄ = z
r̄zr (61)
regardless of the curved metric under consideration. This is convenient for working
with CHn and CPn, because it makes the SU(n) properties of formulae manifest, and
facilitates the study of the transformation properties under SU(n, 1) and SU(n + 1),
respectively.
The metric on CHn is now defined by:
gµν =
0 grs̄
gr̄s 0
 (62)
where:
grs̄ = gs̄r =
(1− zt̄zt)
zrzs̄
(1− zt̄zt)2
(1− zt̄zt)
δrs̄ +
zrzs̄
(1− zūzu)
so that:
gµνdz
µdzν = grs̄dz
rdzs̄ + gr̄sdz
r̄dzs =
(1− zt̄zt)
δrs̄ +
zrzs̄
(1− zūzu)
dzrdzs̄ (64)
The complex hyperbolic space CHn corresponds to the region zrzr̄ < 1.
We note that gr̄s is the complex conjugate of grs̄, or in other words, gr̄s = (grs̄)
In general, when working with complex coordinates, I shall choose definitions in ac-
cordance with a convention such that if every index of a vector, tensor, or matrix is
of definite holonomic type, i.e. either holonomic or antiholonomic, but not an index,
such as µ in this section, which can be either, then replacing every unbarred index by
the corresponding barred index, and every barred index by the corresponding unbarred
index, is equivalent to complex conjugation.
From (62) and (63) we find:
(1− zt̄zt)2n+2
(1− zt̄zt)d+2
where g denotes, as usual, the determinant of gµν , and d = 2n. Also:
gµν =
0 grs̄
gr̄s 0
 (66)
where
grs̄ = gs̄r = (1− zv̄zv) (δrs̄ − zrzs̄) (67)
We observe that
grs̄ = −∂r∂s̄ ln
1− zt̄zt
so the metric is Kähler, with Kähler potential − ln
1− zt̄zt
. The Kähler form is
ωrs̄ = −igrs̄, ωs̄r = igrs̄ (69)
and has real components in a real coordinate system. The nonvanishing Christoffel
symbols of the second kind are:
Γrst =
δr tzs + δ
(1− zv̄zv)
, Γr̄s̄t̄ =
δr̄ t̄zs̄ + δ
s̄zt̄
(1− zv̄zv)
Hence, recalling the sign convention (6) for the Riemann tensor, we have:
R urs̄t = −∂s̄Γurt = −δutgrs̄ − δurgts̄, R ūrs̄t̄ = ∂rΓūs̄t̄ = δūt̄grs̄ + δūs̄grt̄ (71)
Rrs̄tū = −grs̄gtū − grūgts̄ (72)
and the Ricci tensor:
Rrs̄ = (n+ 1) grs̄ (73)
so the metric is Kähler-Einstein.
To calculate the quantity ǫν1...ν2nǫ
µ1...µ2nR ν1ν2µ1µ2 . . . R
ν2n−1ν2n
µ2n−1µ2n
, which occurs
in the generalized Gauss-Bonnet formula [122], we note that, if we define the ten-
sor, gν1...ν2n , by gν1...ν2n ≡
gǫν1...ν2n , then on transforming to complex coordinates, as
described after (54), gν1...ν2n becomes det V gν1...ν2n . I shall assume that the complex
coordinates are taken in the order 1, 1̄, 2, 2̄, 3, 3̄, . . ., so that V is block diagonal. Then
det V = in. Thus g11̄22̄33̄...nn̄ = i
n√gǫ123...nǫ1̄2̄3̄...n̄, so ǫ11̄22̄33̄...nn̄ = inǫ123...nǫ1̄2̄3̄...n̄. We
then find, from (72), that:
ǫν1...ν2nǫ
µ1...µ2nR ν1ν2µ1µ2 . . . R
ν2n−1ν2n
µ2n−1µ2n
= 22nn! (n+ 1) ! (74)
If we now formally introduce an (n+ 1)th coordinate, zn+1, that is actually set
equal to 1, so that dzn+1 = 0, let indices R, S, . . ., run from 1 to n + 1, and define
ηRS̄ = (δrs̄,−1), then the above formula (64), for the squared line element of CHn, can
be written:
gµνdz
µdzν =
−ηT̄ Y zT̄ zY
ηRS̄dz
RdzS̄ +
zX̄ηRX̄dz
zV ηS̄V dz
−ηŪWzŪzW
= − 2
ηT̄ Y z
T̄ zY
dzR − zR ηZX̄z
X̄dzZ
ηB̄Cz
dzS̄ − zS̄ ηĀV z
V dzĀ
ηŪW z
which, apart from the overall minus sign, and the factor of 2, which results from
my choice of normalization in (63), would become equal to the Fubini-Study metric on
CPn, as in equation (15.3.14) of [72], if each ηDĒ was now replaced by δDĒ. Conversely,
if we now relax the condition zn+1 = 1, (75) can be regarded as the metric on CHn in
homogeneous coordinates, since, just as with the Fubini-Study metric in homogeneous
coordinates, (75) is invariant under rescaling of the zR, and also vanishes if dzR is a
multiple of zR, or dzS̄ is a multiple of zS̄. Indeed, just as in the case of the Fubini-
Study metric, (75) is invariant under holomorphic position-dependent rescalings of the
coordinates: zR → f (z) zR. For (75) is the square of the distance between z, and a
nearby point z′ = z + dz. And zR → f (z) zR implies:
R → f (z′) (z′)R = f (z) zR + f (z) dzR + (∂Df (z))
zR (76)
Hence dzR → f (z) dzR+(∂Df (z))
zR, and the second term here cancels because
it is a multiple of zR. CHn corresponds to the region, in the space of the homogeneous
coordinates, such that
ηRS̄z
RzS̄ < 0 (77)
Thus zn+1 does not vanish anywhere on CHn, so, unlike the case of CPn, CHn is
naturally covered by a single coordinate patch.
The metric (75) is manifestly invariant under linear SU(n, 1) transformations of the
homogeneous coordinates:
zR → LRSzS = LRszs + LRn+1zn+1 (78)
where the matrix LRS satisfies:
ηRS̄ = ηT ŪL
S̄ (79)
so that ηRS̄z
RzS̄ is invariant. The matrix LŪ
in (79) is by definition equal to
i.e. the complex conjugate of LUS, in accordance with the convention stated above.
Now the hypersurface zn+1 = 1, in the space of the homogeneous coordinates, is
equivalent to the whole of CHn, in the original coordinates. The action of (78), on
points in this hypersurface, is:
zr → Lr szs + Lr n+1
zn+1 → Ln+1 szs + Ln+1n+1 (80)
Thus this hypersurface is not, in general, left invariant by (78). However, as noted
above, (75) is also invariant under holomorphic position-dependent rescalings of the
coordinates: zR → f (z) zR. Hence (75) is invariant under (78), followed by division
by the new value of zn+1. This compound transformation leaves the hypersurface
zn+1 = 1 invariant, and transforms the points in this hypersurface, which correspond
to the points of CHn, by the projective transformations:
Lr sz
s + Lr n+1
Ln+1 sz
s + Ln+1n+1
Thus, since the original squared line element, (64), is the restriction of (75) to the
hypersurface zn+1 = 1, the original squared line element, (64), is also invariant under
the projective SU(n, 1) transformations, (81). This can also be verified directly.
Returning now to the original coordinate system, or in other words, setting zn+1 =
1, and restricting z to represent the n-vector, (z1, . . . , zn), we can send the origin of
CHn to an arbitrary point, z, of CHn, by means of an SU(n, 1) projective transforma-
tion, (81), by choosing the matrix LRS to be the SU(n, 1) “boost”:
LRS =
Lr s L
Ln+1 s L
δrs +
ztzt̄
zrzs γz
γzs γ
 (82)
where
1− ztzt̄
For n ≥ 2, CHn is not maximally symmetric, and the sectional curvature is not
constant. In general, for linearly independent vectors Aµ, Bν , the sectional curvature,
at a point of a Riemannian manifold, is defined, bearing in mind the sign convention
(5), by:
K (S) = − RµνστA
µBνAσBτ
(gµσgντ − gµτgνσ)AµBνAσBτ
where S denotes the linear space spanned by Aµ and Bν . To apply this to CHn, in the
complex coordinates, we note that, if a vector, Aµ, is real, in real coordinates, then after
transforming to complex coordinates, as described after (54), its components satisfy
Ar̄ = (Ar)
, just as for the complex coordinates themselves. For the CHn metric, (62),
(63), with the Riemann tensor components (72), we find:
RµνστA
µBνAσBτ =
gtr̄A
(gsūB
sBū)+2
gtūA
(gsr̄B
sAr̄)−2 (grs̄ArB s̄)
2−2 (gsr̄BsAr̄)
(gµσgντ − gµτgνσ)AµBνAσBτ =
= 4 (grs̄A
rAs̄)
gtūB
− 2 (grs̄ArB s̄)
gtūB
− (gsr̄BsAr̄)
2 − (grs̄ArB s̄)
If we now work at the origin of the complex coordinates, so that grs̄ = δrs̄, we can
define real magnitudes |A|, |B|, and real angles, θ, ϕ, by:
|A| =
grs̄ArAs̄, |B| =
grs̄BrB s̄,
θ = arccos
(grs̄ArB s̄) (gtūBtAū)
|A| |B|
arcsin
grs̄A
rB s̄
gtūBtAū
− gtūB
grs̄ArB s̄
Thus:
RµνστA
µBνAσBτ
(gµσgντ − gµτgνσ)AµBνAσBτ
(1 + cos2 θ (1− 2 cos 2ϕ))
(2− cos2 θ (1 + cos 2ϕ))
Now, if n = 1, the angle θ is 0, so the sectional curvature is −2. For n ≥ 2, and for all
values of ϕ, such that cos 2ϕ 6= 1, the right-hand side of (88) varies between a minimum
of −2, when cos2 θ = 1, and a maximum of −1
, when cos2 θ = 0, since if we replace
cos2 θ by x, the function has no maximum or minimum between x = 0 and x = 1. For
cos 2ϕ = 1, the right-hand side of (88) is equal to −1
, except for cos2 θ = 1. And when
cos 2ϕ and cos2 θ are both equal to 1, Aµ and Bν are no longer linearly independent.
Thus for n ≥ 2, the sectional curvature of CHn, with the metric (62), (63), lies in the
range −2 to −1
The equation for a geodesic is:
+ Γrst
(1− zv̄zv)
= 0 (89)
For the geodesics through the origin, we have zr = Zr tanh (αs), where Zr is a fixed
complex n-vector such that ZsZ
s = 1, and α is a real constant. If we choose α = 1√
then s is the geodesic distance from the origin to z, in accordance with (64). Hence
the geodesic distance from the origin to z, is
1 + |z|
1− |z|
where |z| ≡
zrzr. To find the geodesic distance from a point, z, to a point, w, we
can use the invariance of the geometry under the projective SU(n, 1) transformations
(81). Sending z to the origin, by the inverse of (82), sends w to a point w̃, such that:
w̃rw̃
(wr − zr) (wr − zr)− zrzrwsws + wrzrzsws
(1− ztwt) (1− wuzu)
Hence:
cosh2
1− w̃rw̃r
(1− zrwr) (1− wszs)
(1− ztzt) (1− wuwu)
where s now denotes the geodesic distance from z to w. Thus in the homogeneous
coordinates, we have:
∣ηRS̄z
∣ηT Ūz
T zŪ
∣ηV X̄w
where |·| denotes the absolute value of a complex number.
2.3 The field equations and boundary conditions
I shall now assume that Hořava-Witten theory has been quantized in accordance with
standard procedures for quantizing supergravity in eleven dimensions [123, 124], to-
gether with an appropriate treatment of the orbifold fixed-point hyperplanes, and seek
solutions of the field equations, and boundary conditions, that follow from varying the
quantum effective action, or in other words, the generating functional of the proper
vertices, Γ, [125, 126, 87, 88], with respect to the fields. The quantum effective action
is expanded in terms of the number of loops in Feynman diagrams, which in the bulk,
is an expansion in powers of κ2, and on the orbifold fixed-point hyperplanes, is an
expansion in powers of λ2, where λ and κ are related by (45). Since κ and λ are di-
mensional constants, the actual expansion parameters have the form κ
, and λ
, where
L will be, in general, the smallest physically relevant distance, in a particular region
of the geometry. I shall seek solutions with a “thick pipe” form of geometry, so that,
in particular, if L denotes the geodesic distance between the two orbifold fixed-point
hyperplanes, then κ
≪ 1. Furthermore, if L denotes a radius of curvature of the
compact six-manifold, which will in general be either smaller than, or comparable to,
its diameter, on account of the hyperbolic nature of the manifold, then we will again
have κ
≪ 1, throughout the main part of the bulk.
Thus, throughout the main part of the bulk, it will be a good approximation to
neglect all quantum corrections to Γ, and approximate Γ as the gauge-fixed classi-
cal action, together with the Fadeev-Popov terms. We then seek a solution, of the
field equations that follow from varying Γ with respect to the fields, in which all the
Fadeev-Popov fields, and also any other fields introduced in the course of the gauge-
fixing, vanish. The field equations then reduce to the classical field equations for the
supergravity multiplet, which are the Euler-Lagrange equations for the Cremmer-Julia-
Scherk action (25), together with gauge-fixing conditions. We can always solve such
equations by solving the Cremmer-Julia-Scherk field equations in any convenient gauge
we choose, then applying gauge transformations to the solution, in order to satisfy the
required gauge conditions.
I shall now denote the full eleven-dimensional metric by GIJ . It will be distin-
guished from GIJKL by context, and the number of indices. Other conventions are
as in Subsection 2.1, on page 17. Furthermore, coordinate indices A,B,C, . . . will be
tangent to the compact six-manifold, and coordinate indices µ, ν, σ, . . . will be tangent
to the four observed space-time dimensions, which at the inner surface of the thick
pipe, where we live, in this type of model, are the extended dimensions. I shall use the
gauge freedom of general coordinate invariance, in order to choose Gaussian normal
coordinates, such that Gyy = 1, and GUy = 0, and thus seek a solution where the
metric has the form:
ds211 = GIJdx
IdxJ = a (y)
gµνdx
µdxν + b (y)
hABdx
AdxB + dy2 (94)
where gµν is the metric on a four-dimensional locally de Sitter space, whose de Sitter
radius I shall set equal to 1, and whose spatial sections may have been compactified,
as discussed after equation (22), and hAB is the metric on a smooth compact quotient
of CH3, and is locally equal to the metric specified in (62) and (63). Thus Rµν (g) =
−3gµν , and RAB (h) = 4hAB. I shall also consider the possibility of flat and AdS
spacetimes, which would have Rµν (g) equal to zero, and a positive multiple of gµν ,
respectively.
I shall seek solutions such that the inner and outer surfaces of the thick pipe, or, in
other words, the orbifold fixed point hyperplanes, are at y = y1 and y = y2, where y1
and y2 are determined by the boundary conditions, and are independent of position in
the four observed dimensions, and on the compact six-manifold. We are free to shift
y by a constant, and I shall use this freedom to obtain the simplest formulae for the
solution in the bulk, rather than to set y1 or y2 to any particular value.
The inner surface of the thick pipe, where we live, will be at y = y1, so in the de Sit-
ter case, it follows from (22), that we require a (y1) = 16.0 Gyr = 1.51×1026 metres =
0.94× 1061
2.3.1 The Christoffel symbols, Riemann tensor, and Ricci tensor
The non-vanishing Christoffel symbols of the second kind, for the metric ansatz (94),
Γµνσ =
gµτ (∂νgστ + ∂σgντ − ∂τgνσ)
Γµνy = Γ
Γyµν = −aȧgµν = −
ΓABC =
hAD (∂BhCD + ∂ChBD − ∂DhBC)
ΓABy = Γ
AB = −bḃhAB = −
GAB (95)
where a dot denotes differentiation with respect to y. From this, and the formula
(6), for the components of the Riemann tensor, it follows that the only non-vanishing
components of the form R JUV I , of the Riemann tensor, in eleven dimensions, are of the
forms R τµνσ , R
ABC , R
µAν , R
Aµν , R
µAB , and R
AµB . In particular, neither I, nor
J , can be y. The non-vanishing components of the Riemann tensor, when one or more
of the indices is y, are R yµyν , R
yµν , R
µyy , R
yµy , R
AyB , R
yAB , R
Ayy , and R
yAy .
We find:
R τµνσ = R
µνσ (g) +
ν − Gνσδ τµ
R DABC = R
ABC (h) +
B −GBCδ DA
R BµAν =
A , R
AµB =
R yµyν =
Gµν , R
yµy =
AyB =
GAB, R
yAy =
δ BA (96)
where R τµνσ (g) denotes the Riemann tensor calculated from the four-dimensional met-
ric gµν , and R
ABC (h) denotes the Riemann tensor calculated from the six-dimensional
metric hAB. From (96), we find that the non-vanishing Ricci tensor components, in
eleven dimensions, are:
Rµν = Rµν (g) +
Gµν =
RAB = RAB (h) +
GAB =
Ryy = 4
where I used the relations Rµν (g) = −3gµν , and RAB (h) = 4hAB, from above.
For smooth compact quotients ofH6, we choose the metric hAB forH
6 to have radius
of curvature equal to 1, so that RABCD (h) = hAChBD−hADhBC , and RAB (h) = 5hAB.
Thus for a smooth compact quotient of H6, the term 4
GAB in RAB is replaced by
We also need the Riemann tensor components, on the orbifold fixed-point hyper-
planes, calculated from the ten-dimensional metric, on the orbifold fixed-point hyper-
planes. The ten-dimensional metric is obtained from (94), by setting dy = 0, and
either y = y1, or y = y2, as appropriate. Then M10 is simply the Cartesian product,
of a four dimensional locally de Sitter space, with de Sitter radius a (y1), or a (y2), as
appropriate, and a smooth compact quotient of CH3, with the metric (63) multiplied
by a factor b2 (y1), or b
2 (y2), as appropriate. All the Christoffel symbols and Rie-
mann tensor components with mixed indices now vanish, and the only non-vanishing
Riemann tensor components are now R τµνσ (g) and R
ABC (h).
2.3.2 The Yang-Mills coupling constants in four dimensions
There are inevitably significant Casimir energy density terms in the energy-momen-
tum tensor on and near the inner surface of the thick pipe, due to the Hořava-Witten
relation λ ≃ 5.8κ 23 between the d = 10 Yang-Mills coupling constant λ and κ [2], and
the fact that the d = 4 Yang-Mills coupling constants at unification are not much
smaller than 1, which implies that b1 = b (y1), the value of b at the inner surface of the
thick pipe, is comparable to κ2/9.
The value of b1 is fixed by the value of the Yang-Mills fine structure constants in
four dimensions at unification, αU =
, which will be equal to the value of the QCD
fine structure constant at unification, and the magnitude |χ (M6)| of the Euler number
of the compact six-manifold M6. For by the generalized Gauss-Bonnet theorem [122],
the Euler characteristic, or Euler number, χ (M2n), of an arbitrary smooth 2n-manifold
M2n, is given by:
gǫν1...ν2nǫ
µ1...µ2nR ν1ν2µ1µ2 . . . R
ν2n−1ν2n
µ2n−1µ2n
. (98)
Thus defining V (M6) ≡
h, we find from (74), on page 28, that for a smooth
compact quotient of CH3, with the standard metric (62), (63), as used in the metric
ansatz (94):
= −10.3354χ
And for a smooth compact quotient of H6, with the metric normalized such that
RABCD (h) = hAChBD − hADhBC , as stated after (97), we have:
= −8π
= −16.5367χ
(100)
Then on using (45), and reducing (28) to four dimensions, we find [127] that when
M6 is a smooth compact quotient of CH3:
(4πκ2)
2b61V (M6)
0.2615
|χ (M6)|
(101)
And for a smooth compact quotient of H6, the same relation is obtained, but with the
coefficient 0.2615 replaced by 5
× 0.2615 = 0.1634.
The result (101) depends on the factor 1
, in the definition of tr in (28), cancelling
with a factor 30, in the ratio of the trace of the square of a generator of SU (3), naturally
embedded in E8, in the adjoint of E8, to the trace of the square of the corresponding
generator, in the fundamental representation of SU(3). For standard Grand Unification,
this follows from the corresponding ratio for generators of SO(16), already derived in
subsection 2.1, via the natural embedding SU(3) ⊂ SU(5) ⊂ SO(10) ⊂ SO(16). I will
be using a different chain of natural embeddings in Section 5, namely SU (3) ⊂ SU (9) ⊂
E8, but the embedding of SU(3) in E8, by this chain, is equivalent to the embedding
of SU(3) in E8, by the above SO(16) chain, as follows from, firstly, the equivalence of
the embedding of SU(3) in E8 by the above SO(16) chain, and the embedding of SU(3)
in E8 by the chain SU(3) ⊂ SU(5) ⊂ SO(10) ⊂ E6, secondly, the equivalence of the
embeddings in E8, of all four SU(3)’s, in the chain SU (3)×SU (3)×SU (3)×SU (3) ⊂
E6×SU (3) ⊂ E8, and thirdly, the equivalence of the embeddings in E8, of any three of
the four SU(3)’s in the preceding chain, and the embeddings in E8, of the three SU(3)’s
in the chain SU (3)× SU (3)× SU (3) ⊂ SU(9) ⊂ E8. Thus the required relation also
holds for the embedding of SU(3) in E8, via the subgroup chain I will be using in
Section 5. This result will also be verified directly in Section 5. The subgroup chains
just listed all follow simply by identifying appropriate subsets of the roots of E8, in
the weight diagram of E8, without the need to project the roots to a subspace, and
take linear combinations of roots that coincide after the projection, as required, for
example, for embedding SO(n) into SU(n).
In Section 5, I shall consider E8 vacuum gauge fields, that break E8 to the Standard
Model SU (3)×SU (2)×U (1), in such a way, that the values of the coupling constants,
at unification, are approximately equal to the observed values of the Standard Model
coupling constants, as evolved in the Standard Model, to around 142 to 166 TeV. How-
ever, if κ−
9 is around a TeV, it seems possible that the higher dimensional accelerated
unification mechanism of Dienes, Dudas, and Gherghetta (DDG) [128, 129], might
perhaps reduce the unification energy to not much larger than a TeV, since, provided
|χ (M6)| is not too large, b−11 would also then be not much larger than a TeV. The
supersymmetry in the higher dimensions, required for the DDG mechanism to work,
would of course automatically be present, since it is only the compactification that
breaks the supersymmetry, in the models studied in the present paper. The embed-
ding of SU (3)QCD, in E8, will be equivalent to the usual embedding of SU (3)QCD, in
conventional Grand Unification, as discussed in the preceding paragraph, so, assuming
that the DDG mechanism reduces the unification energy, without altering the unifi-
cation value of the coupling constants, I shall provisionally estimate αU as the value
of the QCD fine structure constant, α3, as evolved to around 142 to 166 TeV, in the
Standard Model, which gives the value:
αU ≃ 0.0602 ≃
(102)
Robinson and Wilczek [12] calculated the one loop gravitational correction to the
renormalization group running of the Standard Model gauge coupling constants in
a four-dimensional framework, and found that, within the region of validity of their
one loop result, the gravitational correction reduces the magnitudes of the Yang-Mills
gauge coupling constants as energies are reached where quantum gravitational effects
become significant. However this result is not directly applicable in the present context,
where higher dimensional effects and quantum gravitational effects become significant
together, so I shall not adjust the provisional estimate (102) for this effect. Pietrykowski
[130] found that the Robinson-Wilczek effect is gauge-dependent, and vanishes in a class
of gauges different from the gauge choice made by Robinson and Wilczek.
Thus when the compact six-manifold is a smooth compact quotient of CH3, we
find:
≃ 1.2772
|χ (M6)|
(103)
And when the compact six-manifold is a smooth compact quotient ofH6, the coefficient
1.2772 is replaced by 1.1809.
2.3.3 The problem of the higher order corrections to Hořava-Witten the-
At the inner surface of the thick pipe, where we live, we must necessarily have λ2 ∼
|χ (M6)| b61, up to factors of order 1, where b1 = b (y1), and χ (M6) is the Euler number
of the compact six-manifold M6, which is an integer ≤ −1. Equivalently, from the
Hořava-Witten relation (45), we must have κ
3 ∼ |χ (M6)| b61, up to factors of order
1. The relation including all factors of order 1 is given in (103), on page 37. This
follows from the fact that the Yang-Mills coupling constant, g, in four dimensions, at
unification, is given by g2 ∼ λ2|χ(M6)|b6
, up to factors of order 1, and g, which is equal to
the QCD coupling constant, at unification, is of order 1. Thus quantum effects must
necessarily be relevant, at the inner surface of the thick pipe, and, from (45), also in
the bulk, near the inner surface of the thick pipe.
One type of quantum effect has already been taken into account, namely the very
existence of the supersymmetric Yang-Mills multiplets, which, as discussed above, are
required to cancel the one-loop chiral anomalies of the gravitinos, which are chiral on
the orbifold fixed-point hyperplanes. In order to consider what other relevant quantum
effects may occur, it is necessary to consider how Hořava-Witten theory is defined,
beyond the long distance limit.
Hořava-Witten theory was formally defined [1, 2] as M -theory on M10 × S1/Z2,
where M -theory is an unknown theory in eleven dimensions, whose defining properties
are that it is the strong coupling limit of type IIA superstring theory [72], and its low
energy limit is supergravity in eleven dimensions. Hořava and Witten suggested that
the theory would have a built in short-distance cutoff, but left open the question of
whether the supermembrane in eleven dimensions [131, 132] would play a role in the
physics of that short-distance cutoff, because at the time, it appeared that, although
the supermembrane contained the states of the Cremmer-Julia-Scherk supergravity
multiplet in eleven dimensions [133], it could not be consistently quantized. The prob-
lem was that, due to supersymmetry, there was no energy cost to deforming the shape
of a membrane by drawing “infinitely thin” tubes out from it, even when the zero point
oscillations of the thickness of the tubes were taken into account, and the spectrum
was therefore continuous [134].
However, the supermembrane has more recently been reinterpreted as a second-
quantized theory [135], the idea being that little bubbles of supermembrane, connected
to one another by infinitely thin tubes, are like independently moving single particles,
with the sums over paths, of the infinitely thin tubes connecting the bubble “parti-
cles”, presumably building up the eleven-dimensional analogue of the static Newtonian
gravitational forces between them. Moreover, from section 12 of [135], it is possible
that the supermembrane mass spectrum (in flat space) corresponds simply to the single
particle and multi-particle states of supergravity.
It is thus possible that the supermembrane is, in fact, a kind of second quantized
version of supergravity in eleven dimensions. If this is correct, there is then no known
physical effect to provide the basis for any difference, at the quantum level, between
M -theory and supergravity, on a smooth background, in eleven uncompactified dimen-
sions, because the classical membrane [136] and five-brane [137] solutions are infinitely
massive, on a smooth background, in eleven uncompactified dimensions, and thus do
not take part in quantum processes.
Now the classical membrane solution of d = 11 supergravity was reinterpreted in
[138] as a sourceless solitonic solution, with the singularity at the origin found in [136]
being reinterpreted as a coordinate singularity at an event horizon, through which the
solution can be continued, although there is a curvature singularity hidden inside the
event horizon. And by an argument of Hull and Townsend [139], involving U -duality
after toroidal compactification to four dimensions, it is known that M-theory cannot
contain a separate fundamental supermembrane, distinct from the solitonic membrane
of d = 11 supergravity. This is consistent with the fact that the full dynamics of type
IIA superstring theory [13] arises from the solitonic membrane of the CJS theory, on
toroidal compactification to ten dimensions. Let us recall how this works [1].
We first recall that, by a generalization of Dirac’s argument [140] for the quantiza-
tion of the product of electric charge and the magnetic charge of a magnetic monopole,
the tensions T2 and T5, of a solitonic membrane and a solitonic fivebrane, are con-
strained quantum mechanically to satisfy [141, 142, 143]:
2κ2T2T5 = 2πn, n ∈ Z (104)
Thus there must be a fundamental membrane tension, that is a numerical multiple of
3 , and a fundamental fivebrane tension, that is a numerical multiple of κ−
3 , such
that the tensions of all solitonic membranes and solitonic fivebranes are constrained
quantum mechanically to be integer multiples of these fundamental tensions. I will
confirm below, without reference to fivebranes, that the fundamental membrane tension
, when the CJS action is 1
M11 d
R + . . .
, as in (25) for Hořava-
Witten theory in the “upstairs” picture [25, 91].
Now since a solitonic membrane in eleven uncompactified dimensions is infinitely
extended and has a nonzero minimum tension, it is infinitely massive, and cannot be
produced in any physical process. Solitonic membranes of finite extent do not exist
quantum mechanically in eleven uncompactified dimensions, because a membrane of
finite extent and tension T2 would contract into a region of size T
2 , the smallest size
allowed by the uncertainty principle, which for T2 not smaller than the fundamental
membrane tension is comparable to or smaller than the thickness ∼ κ 13T
2 of the
membrane [136], so it would look like a lump rather than a membrane. And while
such a lump could exist classically as a black hole, it has the wrong geometry to be a
source of CIJK , so it cannot carry any charge to stabilize it, as an extreme charged state
of nonzero mass, against decay by Hawking radiation [144], so it will not lead to the
existence of any massive single particle states in the spectrum of the uncompactified
CJS theory.
On the other hand, if a solitonic membrane of infinite extent, and tension T2, already
exists in the vacuum, then its effective dynamics, at distances ≫ both κ2/9 and the
thickness ∼ κ 13T
2 of the membrane, can be studied in terms of collective coordinates,
by deriving a worldvolume effective action for the membrane, by the method of Callan,
Harvey, and Strominger [145]. The first step is the same as in studying a Kaluza-Klein
compactification, treating the dimensions parallel to the worldvolume of the membrane
as the “extended” dimensions, and the dimensions perpendicular to the worldvolume
of the membrane as the “compact” dimensions. The CJS fields are decomposed into
blocks according to which of their tensor indices are parallel to or perpendicular to the
membrane, and the spinor index of the gravitino is written as a pair of a two-valued
SO (2, 1) spinor index and a sixteen-valued SO (8) spinor index. Then all the fields
are expanded in terms of a complete set of states on the “compact” dimensions, with
coefficients that depend on position in the “extended” dimensions, or in other words,
on position on the membrane worldvolume.
The membrane thickness ∼ κ 13T
2 now plays the role of the size of the compact
dimensions, and at distances large compared to both κ2/9 and κ
2 , only the massless
modes are dynamically significant. The massless modes correspond to the zero modes
of the solitonic membrane, which have been studied by Kaplan and Michelson [146].
The solitonic membrane is a BPS solution of the CJS theory, so in accordance with
the general analysis of Callan, Harvey, and Strominger, 16 of the 32 supersymmetries
of the CJS theory are realized linearly, as supersymmetries of the world-sheet action,
and the remaining 16 supersymmetries are realized nonlinearly, as massless fermionic
Goldstone modes. Half of the fermionic Goldstone modes vanish on the mass shell,
so there are 8 bosonic Goldstone modes, which are the 8 translational zero modes,
corresponding to translations of the membrane in the 8 directions perpendicular to the
world sheet.
Choosing coordinates such that the membrane is in the (1, 2) plane, let xµ, 0 ≤
µ ≤ 2 denote the coordinates on the worldvolume, and ym, 3 ≤ m ≤ 10 denote
the coordinates perpendicular to the worldvolume. Then an arbitrary diffeomorphism
ym → ym − ξm (y), with infinitesimal parameters ξm (y), generates a zero mode, by:
δgIJ = ξ
K∂KgIJ +
gKJ +
gIK (105)
δCIJK = ξ
L∂LCIJK +
CLJK +
CILK +
CIJL (106)
In the right-hand sides, here, gIJ and CIJK are as given by the classical membrane
solution:
ds2 = Λ−
3ηµνdx
µdxν + Λ
3 δmndy
mdyn (107)
Cµνρ = ±
εµνρΛ
−1 (108)
where
Λ = 1 +
, (109)
ymym, and the membrane thickness, rh, is related to the membrane tension T2 by
r6h =
, where Ω7 is the volume of the unit seven-sphere S
7. The horizon is located
at ρ = 0 in these coordinates.
We note that the metric (107) tends to Minkowski space as ρ → ∞. Let ǫK(i) (y)
be a set of eight linearly independent vector fields in the eight dimensions perpen-
dicular to the membrane, (so ǫ
(i) = 0), such that limρ→∞ ǫ
(i) = δ
i, and such that
when any of the ǫK(i) is used as the diffeomorphism parameter, ξ
K , in (105) and (106),
the corresponding zero modes δgIJ and δCIJK are normalizable, in the sense that
−ggIKgJMδgIJδgKM and
−ggILgJMgKNδCIJKδCLMN are finite, where
the integral extends over the region ρ ≥ 0 outside the horizon. Then when the small
fluctuations of gIJ and CIJK are expanded as
δgIJ =
λ(i) (x) δgIJ
ǫ(i), y
, δCIJK =
λ(i) (x) δCIJK
ǫ(i), y
(110)
where δgIJ
ǫ(i), y
and δCIJK
ǫ(i), y
are given by (105) and (106), respectively, with
ξK taken as ǫK(i) (y), the corresponding change of the CJS Lagrangian, defined as the
integrand of the Hořava-Witten bulk action in the “upstairs” picture, (25), including
−g factor, has been calculated by Kaplan and Michelson [146] as:
δ2L = (111)
(i)∂nǫ
(j) + Λ∂mǫ
(i)∂nǫ
(j) −
(i)∂mǫ
(j) −
(i)∂nǫ
ηµν∂µλ(i)∂νλ(j)
We see that δ2L vanishes for constant ǫm(i), as expected, due to the global translation
invariance of the classical membrane solution, in the directions perpendicular to the
membrane. But constant ǫm(i) do not lead to normalizable modes δgIJ
ǫ(i), y
δCIJK
ǫ(i), y
. Let us try, instead, ǫm(i) = f (ρ) δ
i, where f (ρ) → 0 as ρ → 0, and
f (ρ) → 1 as ρ→ ∞. Then (111) becomes:
δ2L = 1
Λ,ρf∂ρf +
Λ∂ρf∂ρf
Λδij∂ρf∂ρf
ηµν∂µλ(i)∂νλ(j) (112)
Thus after doing the angular integral over the yi, we find:
δ2SCJS = −
∂ρf∂ρf
d3x ηµν∂µλ(i)∂νλ(i) (113)
Thus with this ansatz for the ǫm(i) (y), the coefficient of η
µν∂µλ(i)∂νλ(i) in δ
2SCJS is
nonzero and has the correct sign. Kaplan and Michelson [146] suggest that the uncer-
tainty in the magnitude of the coefficient should be absorbed into the definition of the
λ(i).
Considering, now, the restrictions on the choice of f (ρ) that result from the re-
quirement that the zero modes are normalizable, so that
−ggIKgJMδgIJδgKM
−ggILgJMgKNδCIJKδCLMN are finite, we note that the ρ integrals will cer-
tainly converge at large ρ if f (ρ) tends to 1 rapidly enough as ρ → ∞. While for
ρ→ 0, we find from (105) and (106) that
−ggµσgντδgµνδgστ and
−ggµρgντgσλδCµνσδCρτλ lead to integrals of the form
0 dρρ
3f 2, while
−ggjlgkmδgjkδglm leads to integrals of the forms
0 dρρ
3f 2,
0 dρρ
4f∂ρf , and
0 dρρ
5 (∂ρf)
Thus we can choose f =
for 0 ≤ ρ ≤ L, and f = 1 for ρ ≥ L, where L > 0
and n ≥ 1. Then for L→ 0 and n→ ∞, f (ρ) increases very rapidly from 0 to 1 in a
small interval near ρ = 0, and then stays equal to 1 for all larger ρ. In this limit, we
see from (105) and (106) that λ(i) can be interpreted as X
i, the x-dependent transverse
displacement of the membrane. And with this choice of f (ρ), the integral in (113)
tends to 1
as L→ 0, so for L→ 0 and n→ ∞, we find from (113) that:
δ2SCJS = −
d3x ηµν∂µX
jδij (114)
We now use the fact that the worldbrane effective action is completely determined
by its supersymmetries, up to an overall normalization factor. From the general prin-
ciples discussed in [147], and the fact that the classical solitonic membrane is a BPS
solution that preserves half of the 32 supersymmetries, with the broken supersymme-
tries being realized nonlinearly as Goldstone modes, it follows that the worldbrane
effective action must be the d = 11 supermembrane action found by Bergshoeff, Sez-
gin, and Townsend [131, 132]. However the supermembrane action will be obtained
in “static” gauge, as discussed in section 4 of [132], such that Siegel’s κ symmetry
[148, 149, 150] has been fixed by the gauge choice Xµ = xµ, where XI are the bosonic
coordinates of the supermembrane, and xµ are the coordinates on the worldvolume, as
above.
Reversing the gauge fixing of the κ symmetry, and allowing a general background
satisfying the CJS field equations, the worldvolume effective action of the infinitely
extended classical solitonic membrane solution, that describes its dynamics at distances
large compared to both κ2/9 and the thickness ∼ κ 13T
2 of the solitonic membrane, is
thus the d = 11 supermembrane action of [131]. The bosonic part of the worldvolume
action is then [136, 25]:
−γγµν∂µXI∂νXJGIJ (X) +
2ǫµνσ∂µX
KCIJK (X)
, (115)
where the sign choice in the third term in (115), called the Wess-Zumino term, is the
same as in (108) [136]. Here γµν is the metric on the worldvolume, and XI (x) are
the bosonic coordinates of the supermembrane, so that the Xm (x), in (115), should
correspond to the X i (x) in (114). We note, however, that there is a factor of 1
discrepancy between (114) and (115), which I shall here leave unresolved.
We now note, following [25] and [91], that since CIJK (X) is not gauge-invariant, the
worldvolume action with the bosonic part (115) will not lead to a well-defined quantum
theory, unless changing the gauge of CIJK (X) can only change the worldvolume action
by an integer multiple of 2π. Let us consider a configuration such that XI (x) sweeps
out some closed three-dimensional surface S3, as x sweeps out some region v of the
worldvolume. Then the requirement that the worldvolume action leads to a well-defined
quantum theory implies that if C ′IJK is any gauge transformation of CIJK , the integral
2ǫµνσ∂µX
K (CIJK (X)− C ′IJK (X)) = 3!
(C − C ′)
(116)
must be an integer multiple of 2π. Following [91], we now apply this requirement with
S3 being the equator of a topological 4-sphere S4, such that C and C ′ are the three-
form gauge field on coordinate patches that cover the north and south hemispheres
of S4 respectively. Then from Stokes’s theorem, and recalling that the four-form G
was defined in section 2.1 by GIJKL = 24∂[ICJKL], so that G = 6dC, it follows that√
S4 G must be an integer multiple of 2π.
On the other hand, by an argument of Witten [151], which I review in subsection 2.7,
on page 137, the vanishing of the Pontryagin number of S4 implies that
is an integer. Thus the smallest possible nonzero value of
S4 G is
. The
requirement that
S4 G is an integer multiple of 2π must be satisfied, in particular,
for this value of
S4 G. Hence we find that
, n ∈ Z (117)
which is in agreement with [25], as amended by [91].
Now the above argument for identifying the worldvolume effective action of the
infinitely extended classical solitonic membrane solution, that describes its dynamics
at distances large compared to both κ2/9 and the thickness ∼ κ 13T
2 of the solitonic
membrane, as the d = 11 supermembrane action, also applies for compactification on
S1, when one dimension of the solitonic membrane wraps the S1 and its other dimension
extends infinitely, and for compactification on S1×S1, when the membrane wraps both
S1’s, since these are also BPS solutions of the CJS field equations, that preserve 16 of
the 32 supersymmetries. Furthermore, because these solutions are BPS, it is expected
that the semiclassical quantization of the worldvolume effective action will be exact
[152], and will thus be valid even when the circumference of one or both of the S1’s
is small compared to κ2/9, which is a strong coupling limit for the CJS theory. The
semiclassical quantization of the supermembrane wrapping a two-torus was studied in
[153].
Let us now consider compactification of the CJS theory on an S1 of circumference
L ≪ κ2/9, such that a solitonic membrane of the minimum tension 1
wraps
once around the S1. If we use the same unit of length in ten dimensions as in eleven
dimensions, the solitonic membrane now looks like a string of tension 1
while its thickness is ∼ κ2/9, as in eleven dimensions. The argument above for the
nonexistence quantum mechanically of solitonic membranes of finite extent in eleven
uncompactified dimensions no longer applies, because a string-like solitonic membrane
of finite extent would contract into a region of size ∼
α′ ∼ κ
, the smallest size
allowed by the uncertainty principle, which can be made arbitrarily large compared
to the string thickness ∼ κ2/9, by choosing L sufficiently small compared to κ2/9.
Furthermore, the analogue of the Planck length, in ten dimensions, is
, and
α′ can also be made arbitrarily large compared to this, by choosing L sufficiently
small compared to κ2/9. Thus the worldvolume effective action of the classical solitonic
membrane solution, namely the d = 11 supermembrane action, can be made arbitrarily
accurate at distances comparable to or larger than
α′, by choosing L sufficiently small
compared to κ2/9.
Now the compactification of the d = 11 supermembrane action on S1, when the
membrane also wraps the S1, was calculated by Duff, Howe, Inami, and Stelle [154],
and found to equal the covariant Green-Schwarz action for the superstring [149]. Once
the covariant Green-Schwarz action is obtained, the standard spectrum of the type IIA
superstrings can be obtained in the light cone gauge [13].
However, we still have to take account of the fact that the worldvolume effective
action of the classical solitonic membrane solution has only been related to the d = 11
supermembrane action in the BPS configurations, so each of the two dimensions of
the solitonic membrane is either infinitely extended or wraps a compactification S1.
So, following Hořava and Witten [1], we consider compactification of the CJS theory
on S1 × S1, such that a solitonic membrane of the minimum tension 1
wraps
once around each S1. This is a BPS solution of the CJS theory, so we can choose the
circumference L of one S1 to be ≪ κ2/9. Then the solitonic membrane now looks like
a closed string of tension 1
L, wrapping once around the second S1, whose
radius R we choose to be comparable to or larger than
. Thus we now
obtain the Green-Schwarz action for the closed superstring wrapping once around the
second S1.
Going to light-cone gauge in the limit L≪ κ2/9, the oscillator degrees of freedom of
the closed superstring completely decouple from the wrapping degrees of freedom. If
the large S1 is in the X9 direction, then the bosonic coordinate X9 of the superstring
has the standard expansion:
X9 = x9 + α′p9τ +N9Rσ + i
n 6=0
−in(τ−σ) + α̃9ne
−in(τ+σ)
, (118)
where τ and σ, 0 ≤ σ ≤ 2π, are the timelike and spacelike worldsheet coordinates
of the closed superstring, p9 = M9
, for some integer M9, and N9 is the number of
times the closed superstring wraps the large S1, which is 1 for the configuration in
which we have obtained the closed superstring. The oscillators αnµ and α̃
µ, n 6= 0,
satisfy
αnµ, α
= nδm,−nηµν and
α̃nµ, α̃
= nδm,−nηµν , because the semiclassical
quantization of the BPS solution is exact [152]. And although we have only obtained
(118) in the case when N9 = 1, the fact that the oscillators in (118) are completely
decoupled from the wrapping degrees of freedom shows that these same oscillators
also create the massive single particle states of freely moving superstrings. Applying
the same treatment to the fermionic collective coordinates of the solitonic membrane
solution, we thus obtain all the massive single-particle superstring states of the type
IIA supertring, while the massless single particle states arise from the dimensional
reduction of the d = 11 supergravity multiplet.
Finally, since
α′ ∼ κ
when we measure distances in ten dimensions in the same
units as in eleven dimensions, we should instead use a unit of length in ten dimensions
that is longer than the unit of length used in eleven dimensions by a factor κ
, if we
want to keep
α′, as measured in the new unit of length introduced for ten dimensions,
fixed as κ
→ ∞ with κ fixed. This can be implemented by writing the Kaluza-Klein
ansatz for the d = 11 metric as
ds2 =
gUV dx
UdxV + dy2 (119)
and interpreting the case where gUV = ηUV as ten-dimensional Minkowski space. Here
U and V run from 0 to 9 as in subsection 2.1, and y, the coordinate along the small
S1, runs from 0 to L. The metric gUV in (119) is called the string metric [155], because
it corresponds to choosing a unit of length, in ten dimensions, with respect to which
= (2π2)
9 is independent of L.
Now since the Green-Schwarz action, which describes free superstrings, becomes
exact in the limit L
→ 0 with κ fixed, we expect that the string coupling constant
λ = eφ, where φ is the dilaton, should tend to zero in this limit. This was demonstrated
by Witten [155], by showing that the string metric gUV in (119) is the correct metric
to use for comparison with the low energy effective action of the type IIA superstring,
written in a standard form such that the kinetic terms for the massless fields from the
NS-NS sector include a factor 1
= e−2φ, and the kinetic terms for the massless fields
from the RR sector are independent of the dilaton.
A dynamical dilaton corresponds to the possibility that L, the circumference of
the small S1, can depend on position in the ten large dimensions. To allow for this
possibility, we define L = κ2/9eγ , where γ can depend on xU , and y = L
ỹ, so that
ỹ runs from 0 to κ2/9. The RR vector field AU is the Kaluza-Klein vector field, and
allowing also for a possible nonvanishing AU , the d = 11 metric ansatz (119) becomes:
ds2 = e−γgUV dx
UdxV + e2γ
dỹ −AUdxU
(120)
The massless NS-NS fields are the graviton, the dilaton, and BUV = CUV y, and the
other massless RR field, besides AU , is CUVW . Substituting (120) into the CJS action
(25), the bosonic kinetic terms in the CJS action then become schematically:
R + (∂γ)
+ |dB|2
+ |dA|2 + |dC|2
(121)
where this expression shows only the dependence on κ and γ, not the correct numerical
coefficients of the terms. Comparison with the low energy effective action of the type
IIA superstring, in the standard form described above, then shows that the string
coupling constant is given by λ = eφ ∼ e 32γ = L
, and thus does, indeed, tend to 0 as
L→ 0 with κ fixed.
Thus the full dynamics of type IIA superstring theory is already contained in the
CJS theory of supergravity in eleven dimensions. But since the defining properties of
M-theory are that it is the strong coupling limit of type IIA superstring theory, and its
low energy limit is supergravity in eleven dimensions, there is then no detectable differ-
ence, on a smooth background, between M-theory, and the CJS theory of supergravity
in eleven dimensions.
In section 1.2 of [156], Green, Russo, and Vanhove noted that on compactification of
d = 11 supergravity on a 2-torus of radii rA and rB, terms of the form e
−crB that arise
in the string theory 4-graviton amplitude are not reproduced by Feynman diagrams at
any number of loops. However for the case rB ≪ κ2/9 that they consider, the d = 11
solitonic membrane wrapping rB can form finite mass solitonic closed strings with mass
proportional to rB, that would give terms of this form by propagating as internal lines
of the Feynman diagrams. The contribution of these solitonic closed strings to the
4-graviton amplitude could presumably be calculated, for example, by the collective
coordinate techniques developed by Gervais, Jevicki, and Sakita [157, 158, 159, 160].
Type IIA superstring theory is thought to be UV complete [161, 162, 163, 164,
165, 166, 167, 168], so apart from the factor of 1
discrepancy between (114) and
(115) that I left unresolved, the CJS theory of d = 11 supergravity, with the non-
perturbative effects of the classical membrane and 5-brane solutions properly included
where appropriate, appears to contain the full dynamics of the UV complete type IIA
superstring theory. However the CJS theory has been argued to be UV incomplete
[169, 170, 171, 172], on the basis of the existence of the linearized 4-field counterterms of
dimensions 8, 12, 14, 16, . . . , constructed by Deser and Seminara [173, 174, 175], which
have been proved by Metsaev [176] to be the complete set of linearized 4-field countert-
erms, and the existence of an infinite set of counterterms [15] constructed as integrals
over the full d = 11 superspace [17, 18], together with a 2-loop dimensional regulariza-
tion calculation [169], using the methods developed earlier in [177, 178], which found
that the dimension 20 Deser-Seminara linearized 4-field counterterm would occur with
an infinite coefficient.
Green, Vanhove, Kwon, and Russo have found that the coefficients of some local
counterterms of dimensions ≥ 12 in the d = 11 theory are fixed by calculations in the
type II superstring theories [179, 180, 181, 156], so the paradox of the UV-incomplete
CJS theory containing the full dynamics of the UV-complete type IIA superstring
theory cannot be resolved by ambiguities in the UV completion of the CJS theory,
which would arise as undetermined coefficients of the Deser-Seminara and superspace
counterterms of dimension ≥ 12 in the quantum effective action of the CJS theory,
somehow disappearing during the compactification of the CJS theory on a small circle
to obtain the type IIA theory.
A possible resolution of half of the paradox, that appears to be consistent with all
known results, follows from noting that the Noether completion of the Deser-Seminara
linearized 4-field invariants, to fully non-linear counterterms, invariant under the full
non-linear CJS supersymmetry variations, up to terms which vanish when the CJS field
equations are satisfied, and can thus be cancelled by the addition of higher dimension
terms to the CJS supersymmetry variations [182, 183], has never been carried out,
and with the exception of the unique dimension 8 invariant [176, 24], whose Noether
completion must exist, if M-theory is consistent, because it occurs in the quantum
effective action of d = 11 supergravity with a non-zero coefficient that is fixed by the
tangent bundle anomaly cancellation on five-branes [25, 26, 27, 28, 29], and confirmed
by anomaly cancellation in Hořava-Witten theory [92, 86, 184, 185, 186, 187, 28, 188,
71], and by comparison with types IIA and IIB superstring theory [189, 190], it is
possible that their Noether completions do not exist.
In the case of d = 4, N = 1 supergravity [191, 192], Noether completions were
always found to exist, but this follows from the existence of the auxiliary field formu-
lations [193, 194, 195, 196, 197, 198]. However for the CJS theory in 11 dimensions,
Rivelles and Taylor showed that no similar auxiliary field formulation can exist [199].
An example of an obstruction to Noether completion in 11 dimensions was found by
Nicolai, Townsend, and van Nieuwenhuizen, when they tried to construct an analogue
of the CJS theory using a 6-form gauge field instead of a 3-form gauge field [200].
A possible resolution to the other half of the paradox would be obtained if the can-
didate counterterms constructed as integrals over the full d = 11 superspace [15, 16]
all vanished identically, or alternatively, if an obstruction existed that prevented the
geometrical transformations in superspace [17, 18] from matching the CJS supersym-
metry variations, for a general solution of the CJS field equations, beyond a certain
power of θ. The mapping of the component fields and supersymmetry variations of
a supersymmetric theory into superspace, such that the geometrical transformations
in superspace match the supersymmetry variations of the component fields, is called
gauge completion [201, 202], and for the CJS theory, this was initially carried out only
to leading order in θ [17].
The first terms beyond leading order in the gauge completion mapping of the CJS
theory into superspace were studied by de Wit, Peeters, and Plefka [203], and to
consider whether an obstruction to gauge completion appears in their results, I shall
temporarily adopt their notation. Thus for the following discussion of [203], coordinate
indices µ, ν, ρ, . . . will temporarily run over all eleven bosonic coordinate directions,
r and s are bosonic tangent space indices, and α, β, γ, . . . are fermionic coordinate
indices. The relations between the normalizations of the fields are ψHWµ = 2ψ
CHWµνρ =
CdWPPµνρ , and Gµνρσ =
Fµνρσ.
The first place to look for an obstruction is equation (4.5) of [203], which must
be satisfied by the terms proportional to the supercovariant field strength F̂µνρσ =
4∂[µCνρσ] + 12ψ̄[µΓνρψσ], in the conventions of [203], at order θ
2 in the θ expansion
of the superspace diffeomorphism parameter. Denoting these terms by ǫβNβ
α, where
ǫβ is the parameter of a CJS local supersymmetry variation, that is to be matched
by a combination of superspace diffeomorphisms, local Lorentz transformations, and
possibly also gauge transformations of a superspace three-form superfield, if one is
included, equation (4.5) of [203] reads:
2∂βNγ
θ̄Γµǫ2
νρσλǫ1
F̂νρσλ − (1 ↔ 2) =
(Γrsθ)
ǭ2 (Γ
νρσλ + 24eν
sΓσλ) ǫ1F̂
νρσλ (122)
Here ∂β =
, and Tµ
νρσλ = 1
νρσλ − 8δ[νµ Γρσλ]
. This equation is to be satisfied
for arbitrary local supersymmetry variation parameters ǫ1 and ǫ2, and is thus a three-
index equation for a two-index quantity. Thus it will have no solution, unless the
“source” terms satisfy an appropriate integrability condition. In fact, from the identity:
∂ε (∂βNγ
α + ∂γNβ
α) + ∂β (∂γNε
α + ∂εNγ
α) + ∂γ (∂εNβ
α + ∂βNε
α) = 0, (123)
which follows from the fact that the spinor derivatives anticommute, we find that
integrability of (122) requires that the following expression vanish for arbitrary F̂νρσλ:
(Tµνρσλ)
νρσλ − 2
(Tµνρσλ)
(Tµνρσλ)
νρσλ +
(Γµκ)
Γ0 (Γµκνρσλ + 24gµνgκρΓσλ)
F̂ νρσλ
(Γµκ)
Γ0 (Γµκνρσλ + 24gµνgκρΓσλ)
F̂ νρσλ
(Γµκ)
Γ0 (Γµκνρσλ + 24gµνgκρΓσλ)
F̂ νρσλ (124)
Thus integrability of (122) requires the following expression, which is antisymmetric in
ν, ρ, σ, and λ, and symmetric in ε, β, and γ, to vanish identically:
(Γµνρσλ − 2gµνΓρσλ + 2gµρΓσλν − 2gµσΓλνρ + 2gµλΓνρσ)α γ
(Γµνρσλ − 2gµνΓρσλ + 2gµρΓσλν − 2gµσΓλνρ + 2gµλΓνρσ)α β
(Γµνρσλ − 2gµνΓρσλ + 2gµρΓσλν − 2gµσΓλνρ + 2gµλΓνρσ)α ε
+ (Γµκ)
Γ0 (Γµκνρσλ + 4gµνgκρΓσλ + 4gµσgκλΓνρ + 4gµρgκσΓνλ + 4gµνgκλΓρσ
+4gµσgκνΓρλ + 4gµρgκλΓσν))βγ + (Γ
Γ0 (Γµκνρσλ + 4gµνgκρΓσλ + 4gµσgκλΓνρ
+4gµρgκσΓνλ + 4gµνgκλΓρσ + 4gµσgκνΓρλ + 4gµρgκλΓσν))γε + (Γ
Γ0 (Γµκνρσλ
+4gµνgκρΓσλ + 4gµσgκλΓνρ + 4gµρgκσΓνλ + 4gµνgκλΓρσ + 4gµσgκνΓρλ + 4gµρgκλΓσν))εβ
(125)
This expression (125) is the type of expression that might vanish by a Fierz identity.
To find out whether or not it vanished, I used the fact, reviewed for example in [84],
that for a real representation of the d = 11 Dirac matrices, as assumed here, the 1024
matrices (Γτ1...τn)
α, 0 ≤ n ≤ 5, form a complete basis for real 32 × 32 matrices. We
can therefore find out whether or not (125) vanishes, by contracting it with a general
matrix Xγα, which turns it into an ordinary sum of matrices, with indices εβ or βε,
multiplied, in the case of the first five terms and the last seven terms, by a trace, and
then taking Xγα to be each of these 1024 matrices in turn.
However, due to Lorentz invariance, it is not necessary to take Xγα to be all 1024
of these matrices. Instead, we first note that (125) vanishes by antisymmetry, unless
ν, ρ, σ, and λ are all different. Thus it is sufficient to evaluate (125) for a fixed choice
of ν, ρ, σ, and λ, all different from each other. I chose ν = 0, ρ = 8, σ = 9, and
λ = y, where, as throughout this section, y denotes the tenth spatial direction. We
then find, when we choose Xγα to be a matrix (Γτ1...τn)
α, for any specific value of n,
and any specific values for the indices τ1, τ2, . . . , τn, that each term in (125) is equal to
a coefficient, times either the matrix (Γ0Γκ1...κm)εβ or the matrix (Γ
0Γκ1...κm)βε, where
κ1, . . . , κm are the indices in {ν, ρ, σ, λ}, that are not in {τ1, τ2, . . . , τn}, and the indices
in {τ1, τ2, . . . , τn} that are not in {ν, ρ, σ, λ}, and may be taken in ascending order.
Furthermore, due to the symmetry of (125) in β and ε, the result vanishes automat-
ically, unless m is one of the numbers for which the matrix (Γ0Γκ1...κm)βε is symmetric,
namely 1, 2, 5, 6, 9, and 10. Furthermore, for each value of n, 0 ≤ n ≤ 5, it is sufficient
to consider just one choice of the indices {τ1, τ2, . . . , τn} that gives each of these values
of m, since if the result vanishes for one choice, it will also vanish for any other choice
that gives the same value of m.
I chose X to be the ten matrices Γ1, Γ08, Γ12, Γ89y, Γ12y, Γ189y , Γ123y, Γ1089y , Γ1239y ,
and Γ12345. For each of these ten choices of X , the contraction of (125) with X
α was
found to vanish. The expression (125) is therefore identically zero, so this potential
obstruction to the completion of the gauge completion procedure, at order θ2, in fact
vanishes.
However, this does not yet imply that there is no obstruction to completion of
the gauge completion procedure at order θ2, because the spin-spin components of the
supervielbein also contain terms of order F̂ θ2, and these terms are required to satisfy
equation (4.9) of [203], which is again a three-index equation for a two-index quantity,
and thus will have a nontrivial integrability condition, since it is required to be satisfied
for arbitrary supersymmetry variation parameter ǫ. The “source” terms of equation
(4.9) of [203] include terms similar in structure, although different in detail, from
the source terms in equation (4.5) of [203], reproduced as equation (122) above, and
also a term involving the solution Nβ
α of equation (4.5) of [203], which cannot be
eliminated by use of equation (4.5) of [203], because it does not occur in the combination
(∂γNβ
α + ∂βNγ
α). However Nβ
α does occur in the combination (∂γNβ
α + ∂βNγ
α) in the
integrability condition for equation (4.9) of [203], so that integrability condition could
be checked by substituting for (∂γNβ
α + ∂βNγ
α) from equation (4.5) of [203], without
actually solving equation (4.5) of [203], but I will not do that in this paper.
The evaluation of (125), contracted with each of the ten choices of Xγα listed above,
was speeded up by use of the well-known identities [84]:
ΓµΓν1...νnΓµ = (−1)
(d− 2n) Γν1...νn (126)
ΓµσΓν1...νnΓµσ = −
(d− 2n)2 − d
Γν1...νn (127)
valid in d dimensions. For example, to evaluate the term (Γ0Γµκ089yΓ1239yΓ
, which
arises for the choice X = Γ1239y, we note that we can treat µ and κ here as summed
only over the seven dimensions different from 0, 8, 9, and y. So we split Γ1239y as
Γ123Γ9y and commute the Γ9y to the right, and, with the understanding that µ and κ
are summed only over the range 1 to 7, we also split Γµκ089y as Γ089yΓµκ. We then use
the identity (127) above, with d = 7 and n = 3, to obtain:
Γ0Γµκ089yΓ1239yΓ
(7− 6)2 − 7
Γ0Γ089yΓ123Γ9y
Γ0Γ01238
(128)
We see from above that already at order θ2, the possibility of mapping the CJS
theory into superspace such that the geometrical transformations in superspace match
the CJS supersymmetry variations, for a general solution of the CJS field equations,
requires that nontrivial integrability conditions be satisfied. Thus it is not possible to
conclude, from the construction of a counterterm in standard d = 11 superspace, that
there exists a corresponding higher derivative term, local in the CJS component fields,
whose variation under the CJS supersymmetry transformations is a total derivative
when the CJS field equations are satisfied, without explicitly checking that there are
no nonvanishing obstructions to the gauge completion mapping of the CJS theory into
superspace, up to the highest power of θ that occurs in the superspace counterterm.
For the Duff-Toms superspace counterterms [15] that would be θ32. In the pure spinor
framework of Berkovits [204, 205], there are superspace invariants involving an inte-
gration over only nine components of θ, but it would be necessary to check that there
are no nonvanishing obstructions when the pure spinor constraint is satisfied, at least
through order θ9.
Turning now to the occurrence of fractional powers of κ, in the expansion of the
quantum effective action Γ, Γ is formally given by an expansion in powers of κ2, starting
with the classical action, of order κ−2, followed by the one-loop term, which is formally
independent of κ. However, it is inevitable that other powers of κ will occur, especially
if Γ, which is a non-local functional of the fields, is developed in a low energy expansion,
as a series of local terms, with increasing numbers of derivatives on the fields. Indeed,
the bulk Green-Schwarz term, mentioned at the end of subsection 2.1, which occurs in
such an expansion, is a sum of terms formed from a three-form gauge field, and four
Riemann tensors, with their indices contracted in various ways, using the metric, and
one antisymmetric eleven-tensor, and is of order κ−
3 . As already noted, if there had
been a built-in short distance cutoff, of order κ2/9, then such fractional powers of κ
would have been interpreted as arising from powers of the short distance cutoff. But
we now need to understand where they come from, when there is no short distance
cutoff.
Figure 1 shows a typical Feynman diagram, in the loop expansion of Γ, that can
contribute to the bulk Green-Schwarz term. It has a three-form gauge field prop-
agating in the loop, and the CI1J1K1 vertex comes from the Cremmer-Julia-Scherk
Chern-Simons term in (25), while the ϕIiJi vertices come from the three-form gauge
field kinetic term, with the metric expanded as GIJ = ηIJ +ϕIJ . Each propagator has
two derivatives acting on it, one from the vertex at each end of it, so that, for purposes
of power counting, the line between two neighbouring vertices, say xi and xj , behaves
as |xi − xj |−11. On the basis of power counting, there is a logarithmic divergence when-
ever any n consecutive vertices, such that 2 ≤ n ≤ 4, cluster together, but the position
space integral is in fact conditionally convergent in these regions, and these apparent
divergences, associated with tree subdiagrams, can be dealt with by the method used
ϕI2J2
ϕI3J3ϕI4J4
ϕI5J5
CI1J1K1
Figure 1: A pentagon contributing to the bulk Green-Schwarz term
to prove Theorem 2 of [206]. However, the diagram as a whole has degree of diver-
gence 11, so that, if we choose the three-form gauge field vertex, x1, as the contraction
point of the diagram, then, in the BPHZ framework [207, 208, 209, 210, 211, 212, 213,
214, 215, 216, 217, 218], we have to subtract a counterterm, which, in this instance,
has the form of the “internal function” of the diagram, namely the propagators, with
the derivatives acting on them out of the vertices, times the terms, of degree up to
and including degree 11, of the Taylor expansion of the “external function” of the di-
agram, namely the function CI1J1K1 (x1)ϕI2J2 (x2)ϕI3J3 (x3)ϕI4J4 (x4)ϕI5J5 (x5), about
the point (x1, x2, x3, x4, x5) = (x1, x1, x1, x1, x1). After integrating over x2, x3, x4, and
x5, this counterterm includes terms with the structure of the bulk Green-Schwarz term,
although with a divergent coefficient, as well as many other terms.
Let us now consider a term of degree 11 in this counterterm, which has a total
of eleven derivatives acting on ϕI2J2 (x1)ϕI3J3 (x1)ϕI4J4 (x1)ϕI5J5 (x1), and a total of
eleven factors, in the counterterm integrand, of the form (xr − x1)Ir , where the index
r runs from 1 to 11, and xr is one of x2, x3, x4, or x5. Suppose we now integrate over
the vertex positions, in the sequence x2, x3, x4, then x5. We see that, when we come
to integrate over the position of x5, the counterterm has an uncancelled logarithmic
divergence at large distances, in consequence of the masslessness of the propagators.
There was no such large distance divergence at all, in the original diagram, if the
classical fields, CIJK , and ϕIJ , are assumed to fall off sufficiently rapidly, at large
distances.
Such large distance divergences, occurring in BPHZ counterterms, but not in the
original diagrams, are a well-known problem of BPHZ renormalization, when there are
massless particles. Traditionally, the problem has beeen dealt with by the BPHZL
method [219, 220, 221, 222], which involves the introduction of regulator masses for
the massless particles, performing additional infra-red subtractions, in addition to the
short-distance subtractions, then letting the regulator masses tend to zero, at the end
of the calculation. An alternative method was presented in [206], where a generalized
BPHZ convergence proof was presented, in Euclidean position space, that allowed the
propagators in the counterterms to differ, at large distances, from the propagators in
the original diagram, without altering the propagators in the uncontracted diagram.
This enables massless propagators, in the counterterms, to be cut off smoothly, at large
distances, so that the large distance divergences are eliminated from the counterterms,
without spoiling the convergence proof, and without altering the propagators in the
uncontracted diagram. The proof in [206] applies only in Euclidean signature position
space, but it seems plausible that Hepp’s convergence proof [214], which can be ap-
plied in Minkowski signature, could be generalized in an analogous way, allowing the
parameter integrals, of the exponentiated propagators, to be cut off at large values
of the exponentiation parameters, in the counterterms, without altering them in the
terms coming from the uncontracted diagram.
When this method is used for a theory such as massless QCD, with no dimensional
parameters in the classical action, the distance at which the smooth long distance
cutoffs of the propagators in the counterterms begin, becomes the distance that provides
the basis for dimensional transmutation [223]. In the case of supergravity in eleven
dimensions, the classical action has precisely one parameter with the dimension of
length, namely κ
9 , so the distance, at which the smooth long distance cutoffs of the
propagators in the counterterms begin, will be a numerical multiple of κ
Now the convergence proof in [206] assumed that the same modified propagators,
differing at long distances from the propagators in the terms coming from the uncon-
tracted diagram, are used in all the terms of the Taylor expansions that occur in the
counterterms, so we will also be using these same modified propagators, with a long
distance cutoff commencing at some fixed numerical multiple of κ
9 , in those terms
in the Taylor expansions in the counterterms, where this is not actually needed, to
ensure convergence at large distances. However, there is not, a priori, any reason to
choose any particular numerical multiple of κ
9 , as the distance at which the smooth
long distance cutoffs of the propagators in the counterterms begin, and if we choose a
different numerical multiple of κ
9 , the result will change by the addition of local finite
counterterms, whose coefficients will involve powers of κ, as determined by dimensional
analysis. In particular, the term with the structure of the bulk Green-Schwarz term,
which contains eight derivatives, will include a factor of κ−
3 . Then, when we require
that the Slavnov-Taylor identities [111, 112, 113], which follow from local supersym-
metry, in the BRST-BV framework [114, 115, 116, 117, 118, 119, 120], are satisfied,
and impose appropriate gauge-fixing conditions, and use the freedom to redefine the
fields, in order to set to zero the coefficients of terms that vanish, when the classical
field equations are satisified, the coefficients of the possible finite counterterms will be
fixed, up to the addition of linear combinations of terms, that correspond to nontrivial
locally supersymmetric higher-derivative deformations, of the CJS theory.
And as I discussed above, it is possible, and consistent with all known results, that
the only non-trivial higher-derivative deformation of the CJS theory, that is locally
supersymmetric at the full non-linear level, might be the deformation whose lowest-
dimension term is the unique dimension-8 CJS on-shell invariant [176, 24] that contains
the bulk Green-Schwarz term.
The numerical coefficient of the unique dimension-8 CJS on-shell invariant [176, 24],
in the quantum effective action of d = 11 supergravity, is fixed by the tangent bundle
anomaly cancellation on five-branes [25, 26, 27, 28, 29], and confirmed by anomaly
cancellation in Hořava-Witten theory [92, 86, 184, 185, 186, 187, 28, 188, 71], and by
comparison with types IIA and IIB superstring theory [189, 190]. This in turn depends
on the Dirac quantization of the two-brane and five-brane tensions [140, 141, 142, 143,
224, 25, 225, 226, 91, 185].
Thus if the unique dimension-8 CJS on-shell invariant is the only non-trivial higher-
derivative CJS on-shell invariant that is locally supersymmetric at the full non-linear
level, it might be possible to calculate the predictions of the CJS theory and Hořava-
Witten theory in the framework of effective field theory, without the occurrence of
undetermined parameters connected with the short distance completion of the theory.
2.3.4 The Casimir energy density corrections to the energy-momentum
tensor
Having now considered some of the problems involved in the definition of Hořava-
Witten theory, or more specifically, the bulk M -theory aspect of it, beyond the long-
wavelength limit, I shall now consider the Casimir-type effects resulting from the com-
pactification on the compact six-manifold.
The Casimir corrections to the energy-momentum tensor, in Einstein’s equations,
arise from the variation of the one loop, and higher loop terms, in the quantum ef-
fective action, Γ, with respect to the classical metric, GIJ . In general, these terms
give corrections to the classical Einstein equations, that are non-local functionals of
the classical metric, GIJ . However, for a given classical metric GIJ , the Casimir terms
in the energy-momentum tensor will be specific functions of position. We can there-
fore adopt an iterative approach to solving the quantum-corrected Einstein equations,
calculating the Casimir terms in a trial classical metric GIJ , then solving the Einstein
equations with these Casimir terms, and if the resulting “output” metric differs from
the “input” metric, repeating the process with an improved “input” metric, until agree-
ment is reached. This method will be used, at the level of rough order of magnitude
estimates, in subsection 2.4.2, on page 93.
The classical metric GIJ will not, in general, be a solution of the classical field
equations, in regions where the Casimir corrections to the energy-momentum tensor
are significant. Nevertheless the gauge-fixed quantum effective action, Γ, is still well
defined, up to possible ultraviolet divergences, as the generating functional of proper
vertices [125, 126, 87, 88]. Moreover it can be calculated, for a classical action A (ϕ),
and for an arbitrary classical field configuration Φ, as the sum of all the one-line-
irreducible vacuum diagrams, calculated from the action A (Φ + ϕ), with the term
linear in ϕ deleted, where ϕ denotes the quantum fields. In other words, using DeWitt’s
compact index notation [227], where a single index, i, runs over all combinations of type
of field, space-time position, and coordinate and other indices, the quantum effective
action, as a function of the classical fields, Φ, is given by the sum of all the one line
irreducible vacuum diagrams, calculated with the action:
A (Φ + ϕ)− ϕi
δA (Φ)
(129)
where the summation convention is applied to the index i. The derivation of this result
is reviewed in section 4, on page 191.
I shall look for solutions such that all physical quantities are covariantly constant
in directions tangential to the four observed dimensions, which is consistent with the
choice of the de Sitter metric for the four observed dimensions, in the metric ansatz
(94). The compactification of CH3 or H6 to the compact six-manifold M6 usually
breaks the homogeneity of the hyperbolic space, so the Casimir terms in the energy-
momentum tensor will not, in general, be covariantly constant in directions tangential
toM6. Furthermore, in the models considered in section 5, on page 222, there are topo-
logically stabilized vacuum Yang-Mills fields on the inner surface of the thick pipe, with
non-vanishing field strengths, whose contributions to the energy-momentum tensor ex-
plicitly break covariant constancy in directions tangential to M6. However, following
Lukas, Ovrut, and Waldram [67], we can introduce a harmonic expansion on the com-
pact six-manifold. I shall work throughout this section at the level of the leading term
in such a harmonic expansion of the energy-momentum tensor, which I shall assume
has the form:
Tµν = t
(1)(y)Gµν , TAB = t
(2)(y)GAB, Tyy = t
(3)(y) (130)
Using the expressions (95), on page 34, for the non-vanishing Christoffel symbols
of the second kind, the conservation equation, DIT
IJ = 0, for the energy-momentum
tensor, now reduces to:
0 = DIT
Iy = ∂yT
Γµµy + Γ
T yy + ΓyµνT
µν + Γ
= ∂yt
(3) +
t(3) − 4 ȧ
t(1) − 6 ḃ
t(2) (131)
We will find that for thick pipe geometries that realize TeV-scale gravity by the
ADD mechanism [3, 5], the energy-momentum tensor, including the contributions of
the four-form field strength GIJKL of the three-form gauge field, is negligible in the
main part of the bulk, well away from the boundaries. Thus the Einstein equations in
the main part of the bulk will, indeed, be consistent with all physical quantities being
covariantly constant on M6. We note that when M6 is a smooth compact quotient of
3, there will be h1,1− 1 additional harmonic (1, 1)-forms on M6 besides the Kähler
form, but only the Kähler form will be covariantly constant. The Kähler moduli do
not correspond to massless modes because, just as for any Kähler-Einstein metric with
a nonvanishing Ricci scalar, each Kähler modulus is equal to a fixed multiple of the
corresponding element of the first Chern class. It seems reasonable to expect that the
effects of the higher harmonics in the Lukas-Ovrut-Waldram harmonic expansion will
decrease rapidly relative to the effects of the leading harmonic, as the distance from
the nearest boundary increases, so that the effects of the higher harmonics will not be
significant, in the main part of the bulk.
The functions t(i) (y) in (130) will be significant near the inner surface of the thick
pipe, where b (y) is ∼ κ2/9. I shall consider three alternative ways in which the outer
surface of the thick pipe might be stabilized, consistent with the observed value (20)
of the effective d = 4 cosmological constant, and in one of the three alternatives, a (y)
is ∼ κ2/9 near the outer surface, so in that case, which is studied in subsection 2.6, on
page 120, the t(i) (y) will also be significant near the outer surface.
To calculate the quantum effective action Γ, and the functions t(i) (y), for a par-
ticular classical metric (94), the propagators and heat kernels for the d = 11 super-
gravity fields, and also for the Fadeev-Popov ghosts [228, 229, 230, 231], the ghosts
for ghosts for the three-form gauge field [232, 233, 123], and possible Nielsen-Kallosh
ghosts [234, 235], are needed for that metric. These can be obtained from the corre-
sponding propagators and heat kernels on an uncompactified CH3 or H6 background,
as appropriate, with the same a (y) and b (y), by the sum over images method of Müller,
Fagundes, and Opher [236, 237, 238], provided the sum over images converges.
For the case of a massless scalar, the sum over images marginally converges when
the action of the massless scalar is as simple as possible, with no “conformal improve-
ment” term, but diverges exponentially, due to the exponential growth of volume with
distance, when a “conformal improvement term” is added to the action, to make the
classical energy-momentum tensor traceless. If the sum over images diverges for any
of the required propagators or heat kernels, it might be possible to obtain the result
by a resummation method [239, 240], or a theta function method [241, 242, 243].
The propagators and heat kernels on a flat R5 times uncompactified CH3 or H6
background can be obtained from the corresponding propagators and heat kernels on
a flat R5 times CP3 or S6 background, which can be calculated by using the Salam-
Strathdee harmonic expansion method [244], and summing the expansions by means of
a generating function. This calculation is currently in progress for CH3, and the scalar
heat kernel on CH3, obtained by this method, is presented in subsection 4.1, on page
196. The leading terms at short distances in the propagators and heat kernels have
been calculated for all the relevant fields on general smooth backgrounds by Burgess
and Hoover [245, 246], using the heat kernel expansion [247, 227]. Casimir effects for
compactification on hyperbolic quotients have also been studied in [248, 249].
Considering, now, the form of the functions t(i) (y) near the inner surface of the
thick pipe, where b ∼ κ2/9, we note that the low energy expansion of the M-theory
quantum effective action, Γ, is known to contain local terms formed from the Rie-
mann tensor and its covariant derivatives. The first such term is formed from four
Riemann tensors, and usually referred to as the t8t8R
4 term [189, 190, 250, 179,
251, 182, 180, 183, 181, 252], where tI1I2J1J2K1K2L1L28 denotes the tensor obtained from
−6gI1J2gJ1I2gK1L2gL1K2 + 24gI1J2gJ1K2gK1L2gL1I2
by antisymmetrizing under I1 ⇀↽ I2,
and symmetrizing under all permutations of (I, J,K, L), with total weight one.
Recalling the definition (14), on page 13, of the energy-momentum tensor, and
looking at the Riemann tensor components (96), on page 34, for the metric ansatz
(94), we see that near the inner surface of the thick pipe, the t8t8R
4 term will result
in terms in the t(i) functions that are numerical multiples of κ
, where the origin of
the non-integer power of κ, in the framework of effective field theory, was explained in
the preceding subsection, and there will also be terms where κ
is multiplied by up
to four powers of ḃ2 or bb̈.
We will find in subsection 2.3.8, on page 77, that the vacuum configurations of
the three-form gauge field CIJK , that result, due to the Hořava-Witten modified
Bianchi identity (42), on page 21, from the presence of topologically stabilized vac-
uum Yang-Mills fields on the Hořava-Witten orbifold hyperplanes, with non-vanishing
field strengths tangential to the compact six-manifoldM6, also produce terms in the t(i)
functions that are numerical multiples of κ
, and in this case, there are no additional
terms involving derivatives of b with respect to y.
Calculations of Casimir energy effects often make use of the proximity force ap-
proximation [253, 254], which in the present case would correspond to treating b as
independent of y, so that all terms with factors of ḃ, b̈, or higher derivatives of b with
respect to y, could be neglected. Thus in this approximation the y direction would
effectively be uncompactified, so that t(3) would be equal to t(1), and t(1) and t(2) would
correspond to d = 11 supergravity on flat R5 times M6. In this case, the first terms
dependent on the topology of M6 would be the one-loop contributions from the terms
in the sum over images other than the identity term. None of these terms contain
short-distance divergences, so their contributions to Γ are independent of κ. The cor-
responding terms in the t(i) functions are thus numerical multiples of 1
Several different indications have been found [189, 190, 250, 179, 180, 181, 255,
256], that suggest that the canonical dimensions of non-vanishing terms, in the low-
energy expansion of the M -theory quantum effective action Γ, in eleven uncompactified
dimensions, will have the form 2 (3k + 1), for integer k, or in other words, 2, 8, 14, . . . .
In that case, the next powers of b, whose coefficients, in the t(i) functions, can get
contributions from local terms in the low-energy expansion of the quantum effective
action, in the context of the proximity force approximation, will be b−14 and b−20.
Neither of these terms would be expected to get contributions at one loop, but both
could get contributions at two loops.
It thus seems reasonable to assume that, within the context of the proximity force
approximation, the functions t(i), near the inner surface of the thick pipe, have expan-
sions of the form:
t(i) = C
+ . . . (132)
where the C(i)n are numerical constants, that depend only on the topology and spin
structure of the compact six-manifold M6, and on the topologically stabilized configu-
rations of the Yang-Mills fields on the Hořava-Witten orbifold hyperplanes, and of the
three-form gauge field CIJK in the bulk.
The conservation equation (131) takes a particularly simple form, near the inner
surface of the thick pipe, when t(3) = t(1), and the t(i) depend only on b, as in the
context of the proximity force approximation. Specifically, when t(3) = t(1):
dt(1)
t(1) − 6
t(2) = 0 (133)
Hence, in this case:
C(2)n = −
(2 + 3n)
C(1)n , C
n = C
n n ≥ 0 (134)
For n = 1, this implies that, in the context of the proximity force approximation, the
topology dependent part of the one-loop Casimir energy-momentum tensor, which is
the b−11 terms in (132), is traceless [257]. This is presumably connected with the formal
relation between the trace of the energy-momentum tensor, and the divergence of the
“dilation current”, and the fact that the b−11 terms in (132) are independent of κ.
The limitations of the proximity force approximation are discussed, for example, in
subsection 4.3 of [258]. In the present case, the proximity force approximation would
not be valid unless ḃ, bb̈, and similar dimensionless quantities formed from b and its
higher derivatives with respect to y, all had magnitude small compared to 1, and we will
find in subsection 2.4, on page 86, that this is not the case. It will therefore be necessary
to go beyond the proximity force approximation, as I will discuss in subsection 2.4.1,
on page 90.
However, for a given trial classical metric GIJ , and in the approximation of dropping
all but the leading terms in the Lukas-Ovrut-Waldram harmonic expansion of TIJ on
M6, we can still assume that the t(i) functions have an expansion of the form (132) near
the inner surface of the thick pipe, provided that b depends monotonically on y in this
region, except that other powers of b, not included in (132), may occur, and we have to
check that when the boundary conditions are satisfied, the “input” t(i) functions lead
self-consistently to a metric that results in “output” t(i) functions equal to the “input”
t(i) functions.
Considering, now, the energy-momentum tensor on the Hořava-Witten orbifold
fixed-point hyperplanes, let T̃ [i]UV , i = 1, 2, be defined by (14), on page 13, with
(SSM + SDM) replaced by the boundary action at y = yi, and the metric gµν replaced by
the induced metric, GUV , on the boundary at y = yi. This is a change of notation from
earlier sections, where the fixed-point hyperplanes were distinguished by a superscript
in round parentheses. Then in the approximation of dropping all but the leading terms
in the Lukas-Ovrut-Waldram harmonic expansions of the T̃ [i]UV on M6, I shall assume
that the T̃
UV have the block diagonal structure:
T̃ [i]µν = t̃
[i](1)Gµν , T̃
AB = t̃
[i](2)GAB (135)
The coefficients t̃[1](j) will receive contributions that are numerical multiples of κ
from the leading terms in the Lukas-Ovrut-Waldram harmonic expansion of the energy-
momentum tensor of topologically stabilized vacuum Yang-Mills fields on the inner
surface of the thick pipe. It would seem reasonable to expect these contributions to be
roughly a positive numerical multiple of the energy-momentum tensor that results from
embedding the spin connection in the gauge group for CH3, which will be calculated
in subsection 2.3.9, on page 80. The Lovelock-Gauss-Bonnet terms in the quantum
effective action on the Hořava-Witten orbifold hyperplanes, discussed in connection
with (48), on page 23, also result in terms in the t̃[1](j) coefficients that are numerical
multiples of κ
, which will also be calculated in subsection 2.3.9.
Thus by analogy with (132), I shall assume that within the context of the proximity
force approximation, the coefficients t̃[1](i) can be expanded as:
t̃[1](i) = D
+ . . . (136)
where the D(i)n are numerical constants, that depend only on the topology and spin
structure of M6, and on the topologically stabilized vacuum configurations of the
Yang-Mills fields on the Hořava-Witten orbifold hyperplanes, and at higher orders, on
the vacuum configuration of the three-form gauge field in the bulk. We note that in
consequence of the Hořava-Witten relation (45), on page 21, between κ, and the Yang-
Mills coupling constant λ on the orbifold hyperplanes, the expansion (136) is equivalent
to an expansion in integer powers of λ.
For the solutions I shall consider in subsection 2.6, on page 120, where a (y) becomes
as small as κ2/9 at the outer surface of the thick pipe, and the three observed spatial
dimensions are assumed to be compactified to a smooth compact quotient M3 of H3,
the expansion analogous to (132) is
t(i) = C̃
+ . . . , (137)
and the expansion analogous to (136) is
t̃[2](i) = D̃
+ . . . . (138)
The situation where a (y) becomes as small as κ
9 at the outer surface differs from
the situation near the inner surface, in that one of the four dimensions scaled by a (y) is
the time dimension, and only the three spatial dimensions scaled by a (y) are assumed
to be compactified. The compactification of the three observed spatial dimensions to
M3 breaks d = 4 Lorentz invariance globally, although not locally, so the Casimir
effects near the outer surface will not, in general, be Lorentz invariant.
Thus for the solutions where a (y) becomes as small as κ2/9 at the outer surface, the
Gµν form of Tµν , in (130), would in general have to be replaced, near the outer surface,
by a more general Robertson-Walker form, and the Gµν form of T̃
µν , in (135), would
also have to be replaced by a Robertson-Walker form. However in subsection 2.6 of this
paper, I shall consider the case where the Casimir effects near the outer surface are, to
sufficient accuracy, consistent with (130) and (135). The coefficients C̃(i)n in (137) and
D̃(i)n in (138) are then numerical constants that depend only on the topology and spin
structure of M3.
2.3.5 The orders of perturbation theory that the terms in the Casimir
energy densities occur at
We recall that in subsection 2.3.3, on page 38, we defined the homogeneity number, of a
local monomial in the CJS fields and their derivatives, to be the number of derivatives,
plus half the number of gravitinos. Let us now extend this definition to an arbitrary
product of the CJS fields and their derivatives, not necessarily all at the same point,
and denote the homogeneity number by h. Then the overall degree of divergence of
an L loop Feynman diagram contributing to a term in the quantum effective action,
or in other words, the generating function of proper vertices, in eleven dimensions,
corresponding to a product of the CJS fields and their derivatives, with homogeneity
number h, is
9b+ 10f + (2− 11) v0 + (1− 11) v1 − 11v2 + 11−N = 9L+ 2− h, (139)
where in the left-hand side of (139), b denotes the number of boson propagators, f
denotes the number of fermion propagators, vq denotes the number of vertices with
2q fermion legs, and N denotes the number of derivatives acting on the CJS fields,
which are here the “background” fields, and we noted that the number of fermion
propagator ends is 2f = 2v1+4v2−F , where F is the number of gravitinos among the
“background” fields, and L = b+ f + 1− v0 − v1 − v2.
The maximum power of κ that can occur for an L loop Feynman diagram contribut-
ing to the quantum effective action, in eleven dimensions, is 2 for each propagator,
minus 2 for each vertex, hence 2 (L− 1). However, as discussed in the second part of
subsection 2.3.3, starting around page 54, when we use BPHZ renormalization, with
propagators in the counterterms that differ from the propagators in the direct terms,
by being cut off at large distances, as allowed by the convergence proofs in [206], so as
to avoid the occurrence of divergences at large distances in the BPHZ counterterms,
due to the presence of massless particles, terms involving lower powers of κ also arise
naturally at L loops.
Specifically, according to the prescription in [206], the same modification of the
propagator, at long distances, is used in all the internal lines of a counterterm part.
Since a unit of distance, namely κ2/9, occurs in the CJS action (25), it is natural to
cut off the propagators, in the counterterms, at distances greater than κ2/9, where
the numerical multiple of κ2/9, at which the cutoff occurs, is likely to get modified
later, in effect, when finite counterterms are added so as to satisfy Slavnov-Taylor
identities. The position-space integral for the BPHZ counterterm that has p extra
derivatives acting on the CJS “background” fields, and contributes to cancelling the
short-distance divergence of a direct term of overall ultraviolet degree of divergence
D, where 0 ≤ p ≤ D, then has the schematic form
∫ κ2/9 xpdx
, where the divergence
at small x cancels against a corresponding ultraviolet divergence in the direct terms.
Thus this integral gives ∼ κ 29 (p−D), which is a power ≤ 0 of κ. The CJS fields, mostly at
separated points in the direct term, are now collected into a local monomial, in the CJS
fields and their derivatives, at a single point, in the counterterm, whose homogeneity
number is hf ≡ h+ p. The total power of κ, including the overall factor κ2(L−1), is
κ2(L−1)κ
(p−D) = κ
9(hf−11), (140)
by (139). This is the correct power of κ to multiply a local monomial, in the CJS fields
and their derivatives, of homogeneity number hf , in order for the quantum effective
action to be dimensionless.
Thus we see that the terms of index n in the expansions (132), namely C(i)n
(n−1)
b8+3n
9 C(i)n
)8+3n
, first occur at a number of loops L, where L is the smallest integer
≥ n+2
, and we also find the corresponding conclusion, for the terms in the expansions
(137). Now for the local terms in the low energy expansion of the quantum effective
action, such as terms formed from products of Riemann tensors, possibly with covariant
derivatives acting on them, and their indices contracted in various ways, a number of
indications have been found that, for at least some terms, their coefficients, which will
be independent of the topology of the background field configuration, do not receive
any further modifications, beyond certain finite orders of perturbation theory [25, 89,
189, 190, 250, 179, 180, 181, 252]. However, for smooth compact quotients of CH3 or
H6, the coefficients in (132) and (137) also receive nonlocal contributions, for example
via the sums over images in the propagators, if these converge, so we would expect
the coefficients C(i)n and C̃
n to receive contributions from all loop orders L such that
L ≥ n+2
Considering, now, the terms of index m in the expansions (136), namely D(i)m
b7+3m
an analogous argument, using power counting as appropriate for Feynman diagrams
in ten dimensions, indicates that D(i)m first receives contributions at a number of loops
L, where L is the smallest integer ≥ m+1
, with a corresponding conclusion, for the
coefficients D̃(i)m in (138). However, Hořava-Witten theory is fundamentally defined in
eleven dimensions, and from the Hořava-Witten relation (45), we see that λ(m−1) is a
numerical multiple of κ
(m−1), so by analogy with the bulk case, it seems likely that the
coefficients D(i)m and D̃
m will, in fact, receive contributions from all loop orders L such
that L ≥ m+2
. For m = −1, this is in agreement with the fact that, in Hořava-Witten
theory, the Yang-Mills actions, on the orbifold ten-manifolds, first arise as a one-loop
effect, while for m ≥ 0, it gives an onset value of L that is less than or equal to that
given by the “d = 10” estimate.
2.3.6 The expansion parameter
Now we found in subsection 2.3.2, on page 35, that for a reasonable estimate, (102),
on page 37, of the d = 4 Yang-Mills fine structure constant at unification, the value
b1 = b (y1) of b (y), at the inner surface of the thick pipe, is related to |χ (M6)|, the
magnitude of the Euler number of the compact six-manifold, by (103), on page 37,
which states that b1
≃ 1.28
|χ(M6)|
, when M6 is a smooth compact quotient of CH3,
and we also find that b1
≃ 1.18
|χ(M6)|
, when M6 is a smooth compact quotient of H6.
Thus to find out whether a particular value of |χ (M6)| is possible, and indeed, whether
|χ (M6)| ≥ 1 is possible, we need to know whether the expansions (132), for the bulk
Casimir energy density coefficients near the inner surface of the thick pipe, and the
expansions (136), for the Casimir energy density coefficients on the inner surface of
the thick pipe, allow b1 to be as small as the value given by (103), for that value of
|χ (M6)|, or whether the expansions (132) and (136) already become infinite, for a
value of b larger than that value of b1.
We assume that the expansion coefficients in (132), from C
1 onwards, and the
expansion coefficients in (136), from D
1 onwards, depend on the topology of M6, and
in particular, that their signs depend on the topology of M6. Kenneth and Klich [259]
and Bachas [260] have recently discovered that Casimir forces are always attractive in
certain circumstances, but their result does not apply in the present context because
M6 has no shape moduli, so that regions of M6 cannot be moved closer together
without also being squeezed at the same time.
Now we know that the Casimir energy densities have local contributions, indepen-
dent of the topology of M6, such as the terms quartic in the Riemann tensor, discussed
in [251, 182, 183], that would contribute terms C
in (132), and the terms on the
boundaries, quadratic in the Riemann tensor, discussed in [67], and mentioned in sub-
section 2.1 above, in connection with equation (48), on page 23, that would contribute
terms D
in (136), which will be calculated in (188), on page 85, when M6 is a
smooth compact quotient ofCH3, and in (192), on page 85, when M6 is a smooth com-
pact quotient ofH6. There will also be local terms built from more covariant derivatives
and powers of the Riemann tensor [189, 190, 250, 179, 180, 181, 252, 255, 256], that
will contribute to the C(i)n and D
n with larger n.
Thus for the phenomenological estimates in this paper, I shall assume that the signs
of the C(i)n and D
n , n ≥ 1, depend on the topology of M6, and that their magnitudes
can depend on the topology of M6 though a factor of order 1, but that, apart from
this factor of order 1, the magnitudes of the C(i)n and D
n , n ≥ 1, are determined by
their typical values, for a geometry of roughly constant curvature. We therefore need
to know what those typical values are.
According to Giudice, Rattazzi, and Wells (GRW) [11], the expansion parameter
for graviton loop corrections in D dimensions, in the sense that perturbation theory
is reliable when the expansion parameter is less than 1, is
2(2π)D
, where
SD−1 =
Γ(D2 )
is the (D − 1)-volume of a unit radius SD−1, E is the relevant energy of
the process, and MD is defined such that the Einstein equation, in D dimensions, is
RAB − 12gABR = −
(2π)(D−4)
TAB. Thus from (25), with
replaced by 2
, so as to work
in the downstairs picture, we find that for Hořava-Witten theory, 2
9M11 = 2π
and the GRW estimate of the expansion parameter for graviton loop corrections is
1890 (2π)
0.0304κ2/9E
(141)
Considering, now, the value of E that would apply for the expansions (132) and
(136), we note, from the discussion after (88), on page 31, that with the metric (62),
(63), the sectional curvature of CH3, at each point of CH3, lies in the range −2 to −1
with the actual value depending on the choice of the two-dimensional section through
the point, so that the magnitude of the corresponding “radius of curvature” lies in the
range 1√
2. Thus when M6 is a smooth compact quotient of CH3, its “radius of
curvature”, at the inner surface of the thick pipe, lies in the range 1√
b1 to
2b1. And
if M6 is a smooth compact quotient of H6, and hAB is in that case normalized so that
RABCD (h) = hAChBD − hADhBC , as assumed after (188), on page 85, then its “radius
of curvature”, at the inner surface of the thick pipe, has the fixed value b1. Thus for
both cases, it is reasonable to take b1 as the typical “radius of curvature” of M6, at
the inner surface of the thick pipe.
Now for the related cases of CP3 and S6, b1 would be the actual radius of curvature,
so the corresponding “wavelength” would be λ = 2πb1, and the corresponding energy
would be E = 2π
. Thus if we also use this estimate of E for the cases of
smooth compact quotients of CH3 and H6, the minimum value of b1
, allowed by the
requirement that the GRW estimate of the expansion parameter be ≤ 1, would be:
≃ 0.03 (142)
which by (103), implies that |χ (M6)| could not be larger than around 6× 109.
On the other hand, since E occurs in the combination E
in (141), it seems possible
that the appropriate value of E should, in fact, be 2π
, in which case the minimum value
of b1
, allowed by the requirement that the GRW estimate of the expansion parameter
be ≤ 1, would be:
≃ 0.2 (143)
which by (103), implies that |χ (M6)| could not be larger than around 7× 104.
As a first check of the GRW estimate of the expansion parameter, we note that,
for D = 4, their estimate of the expansion parameter becomes GN
E2, where GN is
Newton’s constant, (12). Looking now at Donoghue and Torma’s formula for the one-
loop graviton-graviton scattering cross section in D = 4, equation (29) in their paper
[261], and noting their convention for the coupling constant, from their equation (2),
or just after their equations (1) or (2), we see that the expansion parameter is 2GN
where E is the square of the centre of mass energy, times a sum of terms, the first of
which is ln −t
ln −u
, where s, t, and u are the Mandelstam invariants of the scattering
process. Thus in a kinematic region where this sum of terms is ∼ 1, the GRW estimate
of the expansion parameter is, in this case, smaller than the actual parameter, by a
factor ∼ 1
And looking at equation (15) of Donoghue’s calculation of one-loop corrections to
the gravitational scattering of two heavy masses, for D = 4 [262], and noting that
his convention for the coupling constant is the same as Donoghue and Torma’s, we
see that the expansion parameter is GN
|q2|, where q is the momentum transfer, times
a sum of two terms, one of which is −3
ln (−q2), and the other of which, with a
heavy mass in the numerator, is identified, by considering the non-relativistic limit,
as a post-Newtonian correction of classical general relativity, rather than a quantum
correction. Thus, in this case, the expansion parameter for quantum gravitational
corrections is 3GN
|q2| ln (−q2), where −q2 would be multiplied, in the argument of the
logarithm, by an undetermined multiple of GN , that would have to have to be fixed
by an experimental measurement, due to the non-renormalizability of quantum gravity
for D = 4, although it might be determined in a resummation of quantum gravity, for
D = 4, recently developed by Ward [263]. So if we identify |q2| as the GRW E2, we see
that, in the kinematic region where the argument of the logarithm is ∼ 1, the GRW
estimate of the expansion parameter is, in this case, smaller than the actual parameter,
by a factor ∼ 2
. So it appears that, for D = 4, the GRW estimate of the expansion
parameter is reasonable, in kinematic regions where the logarithmic factors it omits
are not too large.
Considering, now, how the GRW estimate of the expansion parameter might be
understood in D dimensions, let us choose the Hořava-Witten downstairs conven-
tion for the gravitational action in D dimensions, so that the Einstein term in the
action is 1
−gR. The GRW estimate of the expansion parameter is then
π2SD−1
(2π)D
. Working, now, in Euclidean signature momentum space, there is
a factor κ
D for each loop, a kinematic factor
for each propagator, where kµ is the
momentum in the propagator, and two numerator momentum factors for each vertex.
Considering, now, a ladder diagram formed from graviton propagators, with an ex-
ternal momentum pµ, with p
2 = E2, running along the ladder, the momentum integral
for each loop of the ladder will be ∼
∫ dDk|k|4
3 , which we would expect to be cut off for |k|
larger than around E, by BPHZ counterterms, and thus to give around
D−2 E
D−2, for
D > 2. Thus, without considering sums over diagrams with a given number of loops,
and the Lorentz index structure of the graviton propagator and vertices, the GRW
estimate of the expansion parameter is obtained for D > 2, up to a factor
(D−2)π2
D−2 .
This factor is 1
for D = 4, so, in view of the two examples above, the estimate so far
is as good as the GRW estimate, for D = 4.
It is not clear, without further investigation, why the magnitude of the Euclidean
loop momentum would tend to be cut off, by BPHZ counterterms, at around E
, rather
than at around E, as suggested by the GRW estimate of the expansion parameter, and
it is also not clear where the extra factor of π2, in the GRW estimate, comes from.
This seems to suggest that, in applying the GRW estimate to the expansions (132) and
(136), E should have been taken as 2π
, resulting in the estimate (143), above, for the
minimum possible value of b1
, rather than the estimate (142), above, except that the
estimate (143), above, could possibly be reduced by a factor of π−
9 , to around 0.15,
with a corresponding increase in the maximum possible value of |χ (M6)|, to around
4× 105.
Considering, now, the effects of sums over diagrams with a given number of loops,
and the Lorentz index structure of propagators and vertices, the fact that the esti-
mate so far includes a factor 1
D−2 , which is absent from the GRW estimate, and is
thus presumably cancelled by Lorentz index contractions, for some diagrams, suggests
considering the limit D → ∞. The D → ∞ limit of the Feynman diagram expansion
of quantum gravity was considered by Strominger [264], and recently reconsidered by
Bjerrum-Bohr [265], and the D → ∞ limit of quantum gravity was also considered, in
the context of a lattice regularization, by Hamber and Williams [266].
The graviton propagator has twoD-vector indices at each end, and includes terms in
which two index-contraction lines run along it, so is in this respect similar to the gluon
propagator at large Nc, when the SU (Nc) adjoint indices, of the gluon propagator,
are written as pairs of an SU (Nc) fundamental index and an SU (Nc) antifundamen-
tal index [267], but that is as far as the similarity with large-Nc Yang-Mills theory
goes. One difference is that the graviton interaction vertices all include two factors of
momentum, and in terms where the D-vector indices of these two momentum factors
are not contracted with each other, a D-vector index line ends on each of them. But
the main difference is that the three-graviton vertex, Vµ1ν1,µ2ν2,µ3ν3, in an expansion
about flat space, includes, in Euclidean signature momentum space, terms of structure
δµ1µ2δν1ν2pµ3qν3 , which allow both index lines from one propagator ending at the vertex,
to pass through the vertex “in parallel”, like a railway track, and leave the vertex to-
gether along another propagator, without getting separated. There are no such terms
in the vertices of SU (Nc) Yang-Mills theory, with its usual action, since they could only
arise from Lagrangian terms with at least two traces, such as tr (FµνFστ ) tr (FµνFστ ).
The presence of such “railway track” terms, in the three-graviton vertex, means
that for some diagrams, there are two factors of D per loop, at large D, and these are
therefore the leading diagrams at large D, so far as index contractions go. For diagrams
built from propagators and three-graviton vertices only, the loops have to be separated
from each other, as one-loop propagator insertions, or one-loop vertex insertions, in
order to be able to have two factors ofD per loop, so in this respect, the large-D limit of
quantum gravity is much simpler than the large-Nc limit of SU (Nc) Yang-Mills theory.
When (2n+m)-graviton vertices, containing terms with n “railway tracks” through
them, with n ≥ 2, m ≥ 1, are included, loops with two factors of D per loop can now
touch one another, and the leading terms at large D, in the quantum effective action,
so far as index contractions go, are “trees” built from one-loop bubbles, that meet one
another at (2n+m)-graviton vertices, n ≥ 2, m ≥ 1, that have n “railway tracks”
through them.
Thus from considering the index contractions from the diagrams that are leading at
large D, so far as index contractions go, the estimate of the expansion parameter now
gets an additional factor D2, so for large D, our estimate of the expansion parameter
is now larger than the GRW estimate, by a factor
D(2π)D−2
. However, for D = 11, the
factor D
is approximately 1, and the factor (2π)
is accomodated by taking E as 2π
in the GRW estimate (141), rather than 1
, as we would initially have expected. Thus
it appears that the second estimate, (143), is at present the best rough estimate of the
minimum value of b1
, and the best rough estimate of the upper bound on |χ (M6)|
is therefore around 7 × 104. To check this estimate further, it would be necessary to
consider diagrams involving the gravitino and the three-form gauge field, but that will
not be done in this paper.
I shall seek solutions of the Casimir energy density corrected field equations and
boundary conditions, such that all the fermionic fields vanish. Thus, in the bulk, the
only non-vanishing fields will be the metric, and the three-form gauge field. I shall now
consider the implications of a topological constraint, that was discussed by Witten in
the context of superstrings, then consider the field equation, and boundary conditions,
for the three-form gauge field.
2.3.7 Witten’s topological constraint
By analogy with a constraint on the compactification of superstrings, discussed by
Witten [45], the fact that the gauge-invariant field strength, GIJKL, is globally well-
defined, implies that for any closed five-dimensional surface, S, we must have
S dG = 0.
If we now work in the “upstairs” picture, so that M11 is M10 × S1, and the fields
transform under reflection in the orbifold fixed-point hyperplanes y = y1, and y = y2,
as discussed in Subsection 2.1, on page 17, and choose S to be the Cartesian product
of the circle S1, and a closed four-dimensional surface, Q, in M10, then this relation,
together with (42), after making the substitutions (47), implies that the sum, over the
two orbifold fixed-point hyperplanes, of the integral:
trF (i) ∧ F (i) −
trR ∧ R
(144)
must be equal to zero. We recall, from the discussion after (28), that for E8, “tr”
denotes 1
of the trace in the adjoint representation, and from the discussion after
(47), that trR[UV RWX] is defined as R
[UV RWX]Y Z , so that the trace is effectively in
the vector representation, of the SO(10) tangent space group of M10.
I will show that, for the metric ansatz (94), the implications of the topological
constraint are the same, regardless of whether the Riemann tensors, in trR[UV RWX],
are calculated entirely in ten dimensions, from the restriction of the metric to the
appropriate orbifold fixed-point hyperplane, or, alternatively, treated as the restriction
to the orbifold fixed-point hyperplane, of the Riemann tensors calculated from the
metric in eleven dimensions.
Now in the problem studied here, M10 is the Cartesian product, of a four dimen-
sional locally de Sitter space, whose three spatial dimensions may have been compact-
ified, and a smooth compact quotient of CH3. Suppose, first, that Q is the Cartesian
product of a topologically non-trivial closed four-dimensional surface, in CH3, and a
point of the locally de Sitter space. And suppose, first, that the Riemann tensors, in
trR[UV RWX], are calculated entirely in ten dimensions, from the restriction of the met-
ric to the appropriate orbifold fixed-point hyperplane. Then
Q trR ∧ R is generically
non-zero, and, moreover, is a topological invariant, specifically a Pontrjagin number,
so both the orbifold fixed points give the same contribution, to the quantity that is
required to vanish. Thus it is necessary to choose nonvanishing E8 vacuum gauge
fields, on one or both of the orbifold fixed-point hyperplanes, in order to cancel the
contributions, to the sum, from
Q trR ∧ R. We recall, from the discussion after (36),
that the trace of the square of a generator of SO(16), in the adjoint representation of
E8, is 30 times the trace of the square of the corresponding generator, in the vector
representation of SO(16). Thus we can satisfy the topological constraint, for all Q of
this type, by choosing one of the two orbifold fixed-point hyperplanes, and choosing
an SO(6) subgroup of the E8 gauge group on that orbifold fixed-point hyperplane, em-
bedded in that E8 gauge group by the natural embedding SO (6) ⊂ SO (16) ⊂ E8, and
setting the E8 Yang-Mills gauge fields, in that SO(6) subgroup of that E8, equal to the
spin connection, while the E8 gauge fields, in the E8 on the other orbifold fixed-point
hyperplane, are zero. Furthermore, the classical Yang-Mills field equation is automat-
ically satisfied for such a configuration, in consequence of the fact that the compact
six-manifold is locally symmetric, so that the covariant derivative of the Riemann ten-
sor, DURVWXY , vanishes identically, which implies that DUR
V W vanishes identically,
where x and y are local Lorentz indices. More generally, the topological constraint,
for this type of Q, will also be satisfied, for arbitrary Yang-Mills field configurations,
in the same topological class, as the configuration just described. This is known as
the standard embedding of the spin connection of the compact six-manifold, in one of
the two E8 gauge groups. In the present case, CH
3 is a Kähler manifold, so the spin
connection will lie in a U(3) subgroup of the SO(6).
Now suppose, instead, that the Riemann tensors, in trR[UV RWX], are treated as the
restriction to the orbifold fixed-point hyperplane, of the Riemann tensors calculated
from the metric in eleven dimensions. In that case, we find, from (96), that:
R JABI R
CDJ = R
ABE R
CDF =
= R FABE (h)R
CDF (h)+4ḃ
2RABDC (h)+2ḃ
4 (hADhBC−hAChBD)(145)
Thus R IJ[AB RCD]IJ = R
[AB RCD]EF = R
[AB (h)RCD]EF (h), so for the metric
ansatz (94), the topological constraint, for a closed four-surface Q, that has the form
of the Cartesian product of a topologically non-trivial four-dimensional closed surface
in the compact six-manifold, and a point in the locally de Sitter space, has exactly
the same form, regardless of whether the Riemann tensors, in trR ∧ R, are calculated
entirely in ten dimensions, from the restriction of the metric to the appropriate orbifold
fixed-point hyperplane, or are the components, in the orbifold fixed-point hyperplane,
of the eleven-dimensional Riemann tensor, and, indeed, it still has the same form, even
if “tr” sums the contracted indices, over all eleven dimensions.
Suppose, now, that Q is the Cartesian product, of a topologically non-trivial n-
dimensional closed surface, in CH3, such that 1 ≤ n ≤ 3, and a topologically non-
trivial (4− n)-dimensional closed surface, in the locally de Sitter space. And as before,
suppose, first, that the Riemann tensors, in trR[UV RWX], are calculated entirely in ten
dimensions, from the restriction of the metric to the appropriate orbifold fixed-point
hyperplane. Then all the Riemann tensor components, with mixed indices, vanish
identically, so
Q trR ∧ R vanishes identically, for any such Q. Furthermore, the E8
vacuum gauge fields already introduced, have no components tangential to the locally
de Sitter space, so
Q trF
(i) ∧ F (i) also vanishes identically, for both E8 gauge groups,
for all such Q. So no nontrivial topological constraint arises from any such Q.
Now suppose, instead, that the Riemann tensors, in trR[UV RWX], are treated as the
restriction to the orbifold fixed-point hyperplane, of the Riemann tensors calculated
from the metric in eleven dimensions. Then it follows from the list, in subsection
2.3.1, of the components of the Riemann tensor, of the form R JUV I , that do not vanish
automatically, for the metric ansatz (94), that
Q trR ∧ R vanishes identically, unless
n = 2. We then find, from (96), that:
R IJµA RνBIJ = R
µA RνBUV = 2R
µA RνBσC = 2
GµνGAB (146)
Hence R IJ[µA RνB]IJ = R
[µA RνB]UV = 0, hence
Q trR∧R vanishes identically, for the
metric ansatz (94), regardless of whether the Riemann tensors, in trR∧R, are calculated
entirely in ten dimensions, from the restriction of the metric to the appropriate orbifold
fixed-point hyperplane, or are the components, in the orbifold fixed-point hyperplane,
of the eleven-dimensional Riemann tensor, and, moreover, this is still true, even if “tr”
sums the contracted indices, over all eleven dimensions.
Finally, there are no topologically non-trivial 4-dimensional closed surfaces, in the
locally de Sitter space, since the time dimension has not been compactified.
Thus Witten’s topological constraint is completely satisfied, by the standard em-
bedding of the spin connection of the compact six-manifold, in the E8 gauge group,
on one of the two orbifold fixed-point hyperplanes, as just described, and this is true,
for the metric ansatz (94), regardless of whether the Riemann tensors, in trR ∧R, are
calculated entirely in ten dimensions, from the restriction of the metric to the appropri-
ate orbifold fixed-point hyperplane, or are the components, in the orbifold fixed-point
hyperplane, of the eleven-dimensional Riemann tensor, and, furthermore, this is still
true, even if “tr” sums the contracted indices, over all eleven dimensions.
The fact that the spin connection is embedded in the E8 gauge group, on just
one of the two orbifold fixed-point hyperplanes, breaks the symmetry between the two
orbifold fixed-point hyperplanes, and it is known from calculations by Witten [127],
and by Lukas, Ovrut, Stelle, and Waldram [68], that when the compact six-manifold
is a Calabi-Yau manifold, the volume of the compact six-manifold is greater, on the
orbifold hyperplane that has the spin connection embedded in its E8 gauge group,
than it is on the other orbifold hyperplane. I will show that this is also true, when the
compact six-manifold is a smooth compact quotient of CH3, so the spin connection
will be embedded in the E8 gauge group, on the outer surface of the thick pipe. This
is fortunate, because we must expect that, in order to find smooth compact quotients
of CH3, such that Fermi-Bose cancellations occur in the Casimir energy densities, to
the precisions required for thick pipe geometries to exist, the Euler number of the
compact quotient will have to be of larger order of magnitude than 1. Thus, if the spin
connection was embedded in the E8 gauge group on the inner surface of the thick pipe,
where we live, the number of generations of chiral fermions would be of larger order of
magnitude than 1, in contradiction with experiment.
In fact, as studied by Pilch and Schellekens [268], the fact that the holonomy group
of the compact six-manifold, in the present case, is SU (3)× U (1), rather than SO(6)
or SU(3), implies that there exist additional ways in which the spin connection could
be embedded in E8×E8, such that the topological constraint is satisfied, and in some
of these ways, part of the U(1) part of the spin connection, is embedded in the E8
on the inner surface of the thick pipe. However, it would seem likely that, for any
uniform embedding of part of the U(1) part of the spin connection, in the E8 on the
inner surface of the thick pipe, in a manner that is independent of position on the
compact six-manifold, the number of chiral fermion modes, on the inner surface of
the thick pipe, would still be comparable, in order of magnitude, to the Euler number
of the compact six-manifold. Thus I shall assume that the entire spin connection is
embedded in the E8 gauge group, on the outer surface of the thick pipe.
In Section 5, on page 222, I shall introduce some E8 vacuum gauge fields, on the
inner surface of the thick pipe, localized on Hodge - de Rham harmonic two-forms, and
partly topologically stabilized by a form of Dirac quantization condition, in order to
break E8 to the Standard Model at around 140 TeV, if the couplings are evolved in the
Standard Model up to unification, and produce a small number of chiral fermions, on
the inner surface of the thick pipe, where we live. This has to be done without spoiling
the satisfaction of Witten’s topological constraint, and I shall also require that, in
the context of Lukas, Ovrut, and Waldram’s harmonic expansion, as discussed above,
the modification to the leading term in each harmonic expansion, resulting from the
introduction of these localized E8 vacuum gauge fields, on the inner surface of the thick
pipe, is a small perturbation of the value which the leading term had, in the absence of
these localized E8 vacuum gauge fields. Thus the analysis of the Casimir energy density
corrected field equations and boundary conditions, in the present Section, should still be
a good first approximation, when the localized E8 vacuum gauge fields are introduced,
in Section 5. The idea is that the compactifications studied in the present Section,
should provide a strong, stiff, robust “platform”, that will only be slightly perturbed,
by the interesting physics of the Standard Model, taking place on the “platform”.
However, it is necessary to note that, since Fermi-Bose cancellations will be required
to take place, to a certain precision, in the leading term in the harmonic expansions of
the Casimir energy density contributions to the energy-momentum tensor, on the inner
surface of the thick pipe, even the small changes to this leading term, resulting from the
introduction of the localized E8 vacuum gauge fields, on the inner surface of the thick
pipe, might imply that a smooth compact quotient of CH3, for which the cancellations
occur to the required precision, in the absence of the localized E8 gauge fields, on
the inner surface of the thick pipe, might have to be replaced by a different smooth
compact quotient of CH3, when those localized E8 gauge fields, are introduced. I will
not be able to determine, in the present paper, whether such a substitution would be
likely to be necessary, and I will simply assume that, if such a substitution is necessary,
then it is made.
If I had chosen the compact six-manifold to be a smooth compact quotient of
the real hyperbolic space H6, rather than of CH3, then RABCD (h) would have been
a constant multiple of (hAChBD − hAChBC), and trR[ABRCD] would have vanished
identically, so that Witten’s topological constraint would not have given any nontrivial
constraints, and there would not have been any need to embed the spin connection in
the gauge group. The symmetry between y1 and y2 would, in that case, have remained
unbroken, at this stage. Nevertheless, we will see that, in this case, thick pipe solutions,
very similar to those obtained for suitable quotients of CH3, will still exist, provided
that the Casimir energy-momentum tensor coefficients satisfy relations similar to those
required for quotients of CH3. The reason for this is that the terms in the energy-
momentum tensor, quadratic in GABCD, the field strength of the three-form gauge
field, as determined by the Hořava-Witten boundary conditions, are only significant,
for the CH3 thick pipe solutions, in at most a very small fraction of the bulk, and, in
fact, at most, only for a small fraction of the region y1 < y < κ
2/9, while for the case
of TeV-scale gravity, we will find that (y2 − y1) ∼ 1015κ2/9. Typical solutions of the
Einstein equations break the symmetry between y1 and y2, even when it is unbroken to
start with, because either b (y) increases monotonically with increasing y, while a (y)
decreases, or vice versa. I always choose the solutions for which b (y) increases with
increasing y, while the warp factor, a (y), decreases, since, by assumption, we live at
y1, with y1 < y2.
Although I mainly consider compactification on quotients of CH3, in this paper,
there are two reasons why compactification on quotients of H6 might turn out to be
preferable. Firstly, on the basis of existing knowledge, the number of smooth compact
quotients of H6, up to a given value of the modulus of the Euler number, might be
very much larger than the number of smooth compact quotients of CH3, up to the
same value of the modulus of the Euler number, as I shall discuss in Section 3, on page
160. And secondly, if the large number of chiral fermion modes, on the outer surface
of the thick pipe, for smooth compact quotients of CH3, should turn out to be a
phenomenological problem, it might be preferable to look for suitable smooth compact
quotients of H6, since there is no need to embed the spin connection of the compact
six-manifold in the gauge group, for smooth compact quotients of H6. However, it
seems possible that the most important criterion, that might favour either CH3 or H6,
is that the local contributions to the coefficients D
1 , in (136), should vanish, if this
is necessary, in order to have an infinite number of smooth compact quotients, with
arbitrarily small, but nonvanishing, values of the D
2.3.8 The field equations and boundary conditions for the three-form
gauge field
When the compact six-manifold is a smooth compact quotient of CH3, we can use
the ansatz of Lukas, Ovrut, Stelle, and Waldram [68], (LOSW), for the four-form field
strength in the bulk, namely that GIJKL vanishes unless all four indices are on the
compact six-manifold, and:
GABCD =
αhABCDEFh
EGhFHωGH (147)
for y1 < y < y2, where α is a fixed number, to be determined by the boundary
conditions, hABCDEF is the tensor
hǫABCDEF , where h is given by (65), and ǫ123456 =
1, and ωGH is the Kähler form, given by (69). This satisfies the Bianchi identities,
and field equations, in the bulk, due to its independence from y, and from position in
the four-dimensional locally de Sitter space, the covariant constancy of hABCDEF , h
and ωFH , and the fact that there are not enough non-vanishing components of GIJKL,
for the GI1...I11G
I4...I7GI8...I11 term in the field equations, to be nonzero. Here GI1...I11
denotes the tensor
−GǫI1...I11 .
To confirm that the vanishing of GIJKL, unless all four indices are on the compact
six-manifold, is consistent with the boundary conditions (43) or (44), after making
the substitutions (47), we recall, from the preceding subsection, that for the metric
ansatz (94), and for all cases of trR[UV RWX], other than trR[µνRστ ], which was not
considered there, the value of trR[UV RWX] = R
[UV RWX]Y Z is the same, regardless
of whether the Riemann tensors are calculated from the restriction of the metric to
the ten-dimensional orbifold hyperplanes, or are taken to be the components on the
orbifold hyperplanes, of the Riemann tensor in eleven dimensions. Furthermore, all
cases of trR[UV RWX] with mixed components vanish identically, and trR[ABRCD] =
R EF[AB (h)RCD]EF (h). For the case of trR[µνRστ ], we find, by a calculation pre-
cisely analogous to the case of trR[ABRCD], that R
[µν Rστ ]IJ = R
[µν Rστ ]ρη =
[µν (g)Rστ ]ρη (g), so, as with the other cases, the result is the same, regardless
of whether the Riemann tensors are calculated from the restriction of the metric to
the ten-dimensional orbifold hyperplanes, or are taken to be the components on the
orbifold hyperplanes, of the Riemann tensor in eleven dimensions, and this remains
true, even if “tr” sums the contracted indices, over all eleven dimensions. However,
the metric gµν is locally de Sitter, specifically dS4, with de Sitter radius equal to 1, so
we have Rµνρη (g) = gµηgνρ − gµρgνη. Hence R ρηµν (g)Rστρη (g) = 2 (gµσgντ − gµτgνσ),
hence R
[µν (g)Rστ ]ρη (g) = 0. Thus the boundary conditions are, indeed, consistent
with the vanishing of GIJKL, unless all four indices are on the compact six-manifold.
To determine α, we note that, in consequence of the decision to embed the spin
connection, of the compact six-manifold, in the E8 at the outer surface of the thick
pipe, it follows from (43), after making the substitutions (47), that near y = y1, we
have:
GABCD =
ǫ (y − y1) trR[ABRCD] + . . . (148)
while near y = y2, we have:
GABCD = −
ǫ (y − y2) trR[ABRCD] + . . . (149)
Thus, setting y = y1+, in (148), and y = y2−, in (149), we see that the boundary
conditions are consistent with GABCD taking the constant value:
GABCD =
trR[ABRCD] =
R EF[AB (h)RCD]EF (h) (150)
for y1 < y < y2.
Now in the complex coordinate system of subsection 2.2, for the compact six-
manifold, we have, from (71), and (72), that:
R EFrs̄ (h)RtūEF (h) = −10hrs̄htū − 2hrūhts̄ (151)
Hence:
R EF[rs̄ (h)R tū]EF (h) = −4 (hrs̄htū − hrūhts̄) (152)
On the other hand, in the complex coordinate system, we have:
hrstūv̄w̄ = i
hǫrstǫūv̄w̄ = 6i
hArstδrūδsv̄δtw̄ (153)
where the factor of i is present because hABCDEF is a tensor, which is real in a real
coordinate system, and on transforming to complex coordinates, for example, by the
matrix U , in (49), h11̄22̄33̄ acquires a factor (detU)
= (−i)−3 = −i, h is given by (65),
with n = 3, and g rewritten as h, and the symbol A, with a list of indices underneath it,
denotes the antisymmetrization of the expression that follows it, under permutations
of those indices. Thus, from (147), we have:
Grst̄ū =
αhrst̄ūEFh
EGhFHωGH =
iαhrst̄ūvw̄h
vx̄hw̄zhx̄z = −
hǫrsvǫt̄ūw̄
hw̄v =
α (hrx̄hsz̄hvq̄ǫxzqǫt̄ūw̄)h
w̄v = −1
α (hrt̄hsū − hrūhst̄) (154)
Comparing with (150), and (152), we see that the ansatz (147) is, indeed, consistent
with the boundary conditions, and that
α = −9
= − 9√
(155)
where I used (46), at the last step. However, it is also interesting to consider compact-
ification on smooth compact quotients of H6, for which there is no need to embed the
spin connection in the gauge group, so that we can set GIJKL = 0. I shall therefore
often leave the above coefficient, α, in the Einstein equations, so that the results for
CH3 can be obtained by setting α = − 9√
, and the results for H6 obtained by
setting α = 0.
Making use of the Kähler geometry identity ωABh
BCωCDh
DE = −δ EA , and the
relation GAB = 1
hAB, we find, from (147), that:
GBFGCGGDHGABCDGEFGH =
GAE (156)
GAEGBFGCGGDHGABCDGEFGH =
(157)
Now, in the upstairs framework, the contribution of the three-form gauge field, to the
energy-momentum tensor, (14), for the bulk action (25), in eleven dimensions, is:
GKNGLOGMPGIKLMGJNOP −
QRGKNGLOGMPGQKLMGRNOP
(158)
Hence the non-vanishing components of T
IJ are:
T (3f)µν = −
6κ2b8
Gµν , T
18κ2b8
GAB, T
yy = −
6κ2b8
(159)
These contributions to the energy-momentum tensor are of the form (134), on page 61,
for n = 0, with C
0 negative. They have been calculated here, for nonzero α, only
for the special case of the standard embedding of the spin connection in the gauge
group on the outer surface of the thick pipe, when the compact six-manifold M6 is
a smooth compact quotient of CH3. However it seems reasonable to expect that in
the approximation of restricting the energy-momentum tensor to the leading term in
the Lukas-Ovrut-Waldram harmonic expansion on M6 [67], as done throughout this
section, the same result would be obtained, but with a different value of α, for the
contributions to the energy-momentum tensor from the vacuum configurations of the
three-form gauge field that result, due to the Hořava-Witten modified Bianchi identity
(42), from the presence of general topologically stabilized vacuum Yang-Mills fields on
the Hořava-Witten orbifold hyperplanes, with non-vanishing field strengths tangential
to M6.
It seems unlikely that the bulk Green-Schwarz term [89, 25] would have a significant
effect, near the inner surface of the thick pipe, because the extended dimensions can
to a good approximation be treated as flat, in this region, and the bulk Green-Schwarz
term includes an antisymmetric tensor, with eleven indices, and the expression con-
tracted with this antisymmetric tensor would, in the approximation that the extended
dimensions are treated as flat, not have any nonvanishing components with enough
different indices, to give a nonvanishing result. I shall assume that the bulk Green-
Schwarz term does not have any significant effect on the field equations of either the
three-form gauge field or the metric, for the geometries considered in the present paper.
2.3.9 The field equations and boundary conditions for the metric
By analogy with (13), the field equations for the gravitational field GIJ , in the upstairs
picture, in eleven dimensions, are:
RIJ −
RGIJ + κ
2TIJ = 0 (160)
where TIJ is now defined by (14), with (SSM + SDM) replaced by the sum of all terms in
the quantum effective action Γ, in the upstairs picture in eleven dimensions, except for
the Ricci scalar term in (25). We note that, due to the incompatibility of a cosmological
constant in eleven dimensions with local supersymmetry in eleven dimensions [200, 269,
124], there is not expected to be any d = 11 cosmological constant term, in the low
energy expansion of Γ.
The Einstein equations (160) can alternatively be written:
RIJ + κ
TIJ −
KLTKL
= 0 (161)
Now subject to the assumptions and approximations discussed in subsections 2.3.4 and
2.3.8, TIJ will have the block diagonal structure (130), on page 58. Thus, using the
Ricci tensor components (97), on page 34, the Einstein equations (161) become:
5t(1) (y)− 6t(2) (y)− t(3) (y)
= 0 (162)
−4t(1) (y) + 3t(2) (y)− t(3) (y)
= 0 (163)
−4t(1) (y)− 6t(2) (y) + 8t(3) (y)
= 0 (164)
where the t(i) (y) satisfy the conservation equation (131), on page 58.
We next need the boundary conditions for the metric, at y1 and y2. Because of the
simple structure of the metric ansatz (94), we can obtain these either directly from the
above Einstein equations, with appropriate delta function terms in the t(i) (y), located
on the orbifold fixed point ten-manifolds, or alternatively, from the Israel matching
conditions [270, 271], which are obtained by including a Gibbons-Hawking term [93, 94]
in the action on the boundary. We recall from subsection 2.1, that Moss’s improved
form of Hořava-Witten theory, which for the purposes of the present paper I assume
to be valid, includes a supersymmetrized Gibbons-Hawking boundary term.
Considering first the direct approach, the energy-momentum tensor T̃
UV , i = 1, 2,
on the Hořava-Witten orbifold hyperplane at y = yi, has the block-diagonal struc-
ture (135), on page 62, by assumption. Hence GKLT̃
KL = 4t̃
[i](1) + 6t̃[i](2). Thus, by
(161) and (130), the first Einstein equation (162) will include delta function terms
5t̃[i](1) − 6t̃[i](2)
δ (y − yi), the second Einstein equation (163) will include delta
function terms κ
−4t̃[i](1) + 3t̃[i](2)
δ (y − yi), and the third Einstein equation (164)
will include delta function terms
4t̃[i](1) + 6t̃[i](2)
δ (y − yi). To match these delta function terms, the slopes of
a (y), and b (y), must be discontinuous, at y1, and y2. Furthermore, by the orbifold
conditions, a (y), and b (y), are to be symmetric, under reflection about y1, and under
reflection about y2. Thus, near y = y1, we must have, for example:
a (y) = a1 + σ |y − y1|+O (y − y1)2 (165)
If we now consider the Einstein equations, (162), (163), and (164), in the vicinity of y1
and y2, and drop all terms except the delta function terms, we find:
5t̃[1](1) − 6t̃[1](2)
δ (y − y1) +
5t̃[2](1) − 6t̃[2](2)
δ (y − y2) = 0 (166)
−4t̃[1](1) + 3t̃[1](2)
δ (y − y1) +
−4t̃[2](1) + 3t̃[2](2)
δ (y − y2) = 0 (167)
4t̃[1](1) + 6t̃[1](2)
δ (y − y1)−
4t̃[2](1) + 6t̃[2](2)
δ (y − y2) = 0 (168)
The third of these three equations follows from the first two, so we only need to
consider the first two. Considering the first equation, near y = y1, we find that
σ, in (165), is given by σ = −κ2
5t̃[1](1) − 6t̃[1](2)
a (y1). Thus we find
y=y1+
5t̃[1](1) − 6t̃[1](2)
. The other boundary conditions follow similarly, and we find:
y=y1+
−5t̃[1](1) + 6t̃[1](2)
y=y1+
4t̃[1](1) − 3t̃[1](2)
(169)
y=y2−
5t̃[2](1) − 6t̃[2](2)
y=y2−
−4t̃[2](1) + 3t̃[2](2)
(170)
Alternatively, we can obtain the boundary conditions from the Israel matching condi-
tions [270, 271], which read:
{KUV −KHUV } = −κ2T̃UV (171)
Here HUV is defined to be the components tangential to the orbifold fixed-point hyper-
plane, of the projection tensor HIJ = GIJ −nInJ , where nI is the unit normal pointing
out of the fixed-point hyperplane, on one side. The curly braces denote summation
over both sides of the fixed-point hyperplane. KUV is the extrinsic curvature of the
fixed-point hyperplane, defined by KUV = H
V DInJ , which is symmetric under
swapping U and V , because nJ will be the gradient of a scalar function, that takes a
fixed value on the fixed-point hyperplane, and whose gradient is normalized, at each
point on the fixed-point hyperplane, so that GIJnInJ = 1 there. K = H
UVKUV . And
T̃UV is the energy-momentum tensor on the fixed-point hyperplane, as above.
In the present case, if we first consider the y = y1+ side of the fixed point hyperplane
at y = y1, we have ny = 1, and all other components of nI vanish, and HUV is simply
the components GUV of GIJ . Furthermore, KUV = −ΓyUV , hence, from (95),
Kµν |y=y1+ =
y=y1+
Gµν , KAB|y=y1+ =
y=y1+
GAB (172)
K|y=y1+ = 4
y=y1+
y=y1+
(173)
At y = y1−, ny = −1, and ȧ and ḃ have also been multiplied by −1, so we recover the
boundary conditions (169), from the Israel matching conditions (171). And we also
recover the boundary conditions (170), in a similar manner.
The energy-momentum tensors T̃
UV , corresponding to the bosonic part of the Yang-
Mills action (28), are given by:
[i]YM
GCDtrF
CDGEF trF
(174)
T̃ [i]YMµν = −
CDGEF trF
DF (175)
Now for compact quotients of CH3, the spin connection has been embedded in the E8
at the outer surface of the thick pipe, while F
AB, and consequently T̃
UV , is zero. And
for compact quotients of H6, the Yang-Mills fields are zero on both surfaces of the thick
pipe, and consequently T̃
UV = 0, for both i = 1 and i = 2.
For the case of CH3, and i = 2, we recall, from subsections 2.1 and 2.3.7, that
for E8, “tr” means
of the trace in the adjoint representation, and that the trace of
the square of a generator of SO(16), in the adjoint representation of E8, is 30 times
the trace of the square of the corresponding generator, in the vector representation of
SO(16). Furthermore, the E8 generators being used, are hermitian. Thus we have:
BD = R
AC RBDEF (176)
I shall now assume that the Riemann tensor R FACE , that is embedded in the E8 on
the outer surface of the thick pipe, is the Riemann tensor R FACE (h), calculated from
the induced metric GUV , on the outer surface of the thick pipe, and not the restriction
to the outer surface of the thick pipe, of the eleven-dimensional Riemann tensor. We
then have R EFAC RBDEF = R
AC (h)RBDEF (h), hence, from (151), we have, in the
complex coordinate system, that:
rs̄ F
tū = −10hrs̄htū − 2hrūhts̄ (177)
GCDtrF
ūD = 16
hrū (178)
We also have GCDtrF
uD = G
CDtrF
ūD = 0. Hence:
GCDtrF
BD = 16
hAB = 16
GAB (179)
T̃ [2]YMµν = −
[2]YM
AB = −
(180)
t̃[2](1)YM = −
, t̃[2](2)YM = −
(181)
When the functions b (y), or a (y), are shown without arguments, they are evaluated
at the appropriate value of y, which for T̃
UV , F
UV , and t̃
[2](i), is at y2.
Now, as discussed in subsection 2.1, the low energy expansion of the quantum
effective action, Γ, on the orbifold fixed point hyperplanes, is believed to contain terms
quadratic in the Riemann tensor, of the Lovelock-Gauss-Bonnet form, obtained from
the Yang-Mills actions (28), by the substitutions (48). The corresponding term in Γ,
at yi, is:
LGB =
−GR [UV[UV R
WX] (182)
I shall now assume, as in the calculation above, of the Yang-Mills energy-momentum
tensor, when the spin connection is embedded in the gauge group, that the Riemann
tensor R XUVW , in (182), is the Riemann tensor calculated from the induced metric
GUV , on the orbifold fixed point ten-manifold M10i , and not the restriction to the
outer surface of the thick pipe, of the eleven-dimensional Riemann tensor. Then, since
M10i is the Cartesian product of the four observed dimensions, and the compact six-
manifold, all the Riemann tensor components, with mixed indices, vanish identically.
The energy-momentum tensors T̃
UV , corresponding to (182), are:
[i]LGB
UV = −
R̃UWXY R̃
V − 2R̃UWVXR̃WX − 2R̃UW R̃ WV + R̃UV R̃
GUV R̃WXY ZR̃
WXY Z +GUV R̃WXR̃
WX − 1
GUV R̃
(183)
where I have now denoted curvatures calculated from the induced metric GUV , on the
orbifold fixed point ten-manifold M10i , by a tilde.
To evaluate (183), we note that when the compact six-manifold is a quotient of
CH3, we have, from (72), that for the metric induced on a fixed point ten-manifold,
by the metric ansatz (94):
R̃ACDER̃
B = 16
GAB, R̃ACBDR̃
CD = 16
GAB (184)
Hence, recalling that gµν , in the metric ansatz (94), is normalized such that Rµν (g) =
−3gµν , and when the compact six-manifold is a quotient of CH3, hAB is normalized
such that RAB (h) = 4hAB, we find, when the compact six-manifold is a quotient of
CH3, that:
R̃WXY ZR̃
WXY Z =
, R̃WXR̃
, R̃ =
(185)
[i]LGB
GAB (186)
T̃ [i]LGBµν =
Gµν (187)
t̃[i](1)LGB =
, t̃[i](2)LGB =
(188)
To evaluate (183) when the compact six-manifold is a quotient of H6, we recall that
in this case we have chosen hAB, in the metric ansatz (94), to be normalized such that
RABCD (h) = hAChBD−hADhBC , so that RAB (h) = 5hAB, as stated after (97), on page
34. We then find, when the compact six-manifold is a quotient of H6, that:
R̃WXY ZR̃
WXY Z =
, R̃WXR̃
, R̃ =
(189)
[i]LGB
GAB (190)
T̃ [i]LGBµν =
Gµν (191)
t̃[i](1)LGB =
, t̃[i](2)LGB =
(192)
Now at the inner surface of the thick pipe, we will have a (y1) ∼ 1026 metres, while
b (y1) will be less than about 10
−19 metres, so for i = 1, we can neglect the terms with
negative powers of a, in (188) and (192). On the other hand, we will find solutions
where a is comparable to b, at the outer surface of the thick pipe, but these solutions
will not be able to fit the observed values of Newton’s constant and the cosmological
constant, and other solutions where a is small compared to b, at the outer surface of the
thick pipe, some of which will be able to fit the observed values of Newton’s constant
and the cosmological constant.
2.4 Analysis of the Einstein equations and the boundary con-
ditions for the metric
The Einstein equations (162), (163), and (164), with the range of y restricted to y1 <
y < y2, together with the boundary conditions (169) and (170), now constitute a system
of coupled ordinary differential equations, and boundary conditions, for the functions
a (y) and b (y).
The functions t(i) (y), defined by (130), on page 58, receive contributions from the
energy-momentum tensor of the three-form gauge field, given by (159) for quotients of
CH3, and 0 for quotients of H6, and from Casimir effects in the bulk, near the inner
surface of the thick pipe, and, for solutions such that a (y) becomes sufficiently small
near the outer surface of the thick pipe, also from Casimir effects in the bulk, near the
outer surface of the thick pipe.
The coefficients t̃[i](j), defined by (135), receive contributions from the energy-
momentum tensor of the Yang-Mills fields on the outer surface of the thick pipe, given
by (181) for quotients of CH3, and 0 for quotients of H6; from the leading terms in the
Lukas-Ovrut-Waldram harmonic expansion, on the compact six-manifold M6, of the
energy-momentum tensor of topologically stabilized vacuum Yang-Mills fields on the
inner surface of the thick pipe; from the Lovelock-Gauss-Bonnet energy-momentum
tensor on the surfaces of the thick pipe, given by (188) for quotients of CH3, and by
(192) for quotients of H6; and from Casimir effects on the inner surface of the thick
pipe, and, for solutions such that a (y) becomes sufficiently small at the outer surface
of the thick pipe, also from Casimir effects on the outer surface of the thick pipe.
The functions t(i) (y), and the coefficients t̃[i](j), are required to be recovered self-
consistently, when they are recalculated for the solution of the Einstein equations and
the boundary conditions.
The equations are invariant under a uniform shift of y, y1, and y2, but as already
noted, in the discussion following (94), I shall use this freedom to obtain the simplest
form of the solution in the bulk, near the inner surface of the thick pipe, rather than
to set y1 or y2 to any particular value.
Eliminating the double derivatives between the three Einstein equations, we find:
κ2t(3) = 0 (193)
When the functions t(i) (y) are shown without arguments, they are evaluated at y.
From (193), we find:
= −2 ḃ
κ2t(3) (194)
The second Einstein equation, (163), now becomes:
− 3 ḃ
± 2 ḃ
κ2t(3) +
−4t(1) + 3t(2) − t(3)
= 0 (195)
Now differentiating (194) with respect to y, we find:
= −2 b̈
− 12 ḃ
− 8 ȧ
κ2ṫ(3)
(196)
where I defined
κ2t(3) (197)
Now, using the formula (196) for ä, the left-hand side of the first Einstein equation,
(162), becomes:
−2± 3ḃ
− 3 ḃ
± 2 ḃ
−4t(1) + 3t(2) − t(3)
κ2t(3)
t(3) − t(1)
−2 ḃ
ṫ(3) +
t(3) − 4
t(1) − 6
(198)
and thus vanishes when (194) and (195) and the conservation equation (131) are sat-
isfied, provided that the square root, (197), is nonvanishing.
Now the third Einstein equation, (164), is equivalent to (194), provided that the
first two Einstein equations are satisfied. Thus (194) and (195), taken together, imply
that all three Einstein equations are satisfied, provided that the conservation equation
(131) is satisfied, and the square root, (197), is nonvanishing. This is true whichever
choice of sign we take in (194) and (195), provided that we choose either the upper
sign in both equations, or the lower sign in both equations.
Now we are seeking solutions in the region y1 ≤ y ≤ y2, such that for y close to y1,
a (y) is very large, and b (y) is very small. Thus we may neglect the term 4
, in the
square root, for y close to y1, in this region. In that case, (195) becomes an ordinary
differential equation for b (y), since, in the approximations discussed above, the t(i) (y)
only depend on y, through b (y), in this region. Moreover, we are looking for solutions
that realize the ADD mechanism [3, 5], by a form of thick pipe geometry, so we require
ḃ > 0, for y greater than y1, and close to y1.
It is convenient to define c (y) ≡ ḃ, so that b̈ = c dc
. Then (195) reduces to a first
order differential equation, for c as a function of b:
κ2t(3) +
−4t(1) + 3t(2) − t(3)
= 0 (199)
We can now carry out a qualitative analysis of the differential equation (199), in the
(b, c) plane. We are interested in the quadrant b > 0, c > 0. For a fixed choice of the
sign of the square root, (199) defines a unique curve through each point in the quadrant
b > 0, c > 0, such that the argument of the square root is non-negative. We can follow
such a curve from the inner surface of the thick pipe, where b is very small.
Suppose, first, we choose the lower sign of the square root, so (199) becomes:
κ2t(3) +
−4t(1) + 3t(2) − t(3)
= 0 (200)
Now the functions t(i) all decrease rapidly in magnitude, with increasing b, and become
negligible as soon as b is large compared to κ2/9. In that case, (200) reduces to:
3c2 − 4
2 (3c2 − 4) (201)
We require c ≥
, in order for the square root to be real. Then dc
≥ 0, and will
typically be ∼ κ− 29 , or larger, once b is ∼ κ2/9. Then once b has increased by a few
multiples of κ2/9, c will be large enough that we can to a reasonable approximation
replace 3c2 − 4 by 3c2, and this becomes a better approximation as c increases further.
Then (201) becomes:
3 + 2
≃ 7.8990
(202)
Thus as soon as b is as large as a few multiples of κ2/9, we have c = db
)7.8990
, for
some constant B, that cannot be much larger than κ2/9, but could be smaller, because
we could be on a trajectory which starts out with a large value of c, near the inner
surface of the thick pipe. Then
≃ 0.7559
(y3−y)
)0.1449
, where y3 is some constant
greater than y1, but such that y3 − y1 cannot be large compared to κ2/9, unless b
somehow remains smaller than B, all the way from y = y1 to y ∼ (y3 −B), which
would require c to be smaller than around κ
(y3−y1) for most of this range. However,
even if the functions t(i) were such that this was possible, and the boundary conditions
could be satisfied, such a solution has no classical bulk, because as soon as a value of
y is reached, such that b is larger than B, b starts increasing very rapidly, and would
reach infinity, if y increased further by more than B. Thus it is not possible to find
solutions with a thick pipe form of geometry, that can realize the ADD mechanism in
a simple way, without considering the upper choice of sign, in (194) and (195).
We now, therefore, choose the upper sign of the square root, so (199) becomes:
κ2t(3) +
−4t(1) + 3t(2) − t(3)
= 0 (203)
We start again, at the inner surface of the thick pipe, where b is very small, and follow
a curve in the (b, c) plane as before, but defined, this time, by (203). The functions
t(i) all become negligible, as before, as soon as b is large compared to κ2/9. Then (203)
reduces to:
3c2 − 4
 (204)
We again require c ≥
, in order for the square root to be real. The right hand
side of (204) is < 0 for all c >
. The simple dependence on b, of the right hand
side of (204), means that the general solution of (204) has the form c = f
, for
some function f , where B is the constant of integration. Thus all trajectories, in this
region, are related to one another, by rescaling b. I shall call the solutions of (204) the
bulk-type trajectories.
Now for c large compared to
, (204) reduces to
≃ −1.8990c
(205)
Thus when b is large compared to κ2/9, and c large compared to
, we have
)1.8990
, (206)
for some constant B. There is now no upper limit to how large B can be, but it cannot
be much smaller than κ2/9. And for large B, this approximate solution will be valid,
throughout the range from b somewhat larger than κ2/9, to b somewhat smaller than
B, and this range of b can be made arbitrarily large, by choosing a sufficiently large
value of B.
The above approximate form (206) of c, as a function of b, corresponds to
≃ 1.4436
y − y0
)0.3449
(207)
for some y0, which we could choose to set to 0, by using the invariance of the equations
and boundary conditions, under a uniform shift of y, y1, and y2.
It is convenient to regard a as a function of b, in the same way as c = db
is being
treated as a function of b. Then in the region where all the t(i) are negligible, the
equation (194) for ȧ
, with the upper choice of sign, and dropping the term 4
in the
square root, becomes:
(208)
When c is sufficiently large, that we are on a bulk type power law trajectory, this
becomes:
= −0.7753a
(209)
Hence:
a = A
)0.7753
(210)
where A is a constant of integration.
Now we will find, in subsection 2.6.1, on page 126, and subsection 2.7, on page 137,
that for TeV-scale gravity, B
, whose value is determined by the boundary conditions
at the inner surface of the thick pipe, is required to have a value around 1.5×10
|χ(M6)|0.1715 .
Thus with our best rough estimate, (143), on page 68, of the minimum value of b1
and the corresponding best rough estimate of the upper bound on |χ (M6)| as around
7×104, we see that B
will be around 104. Thus if the bulk power law, (206), was valid
down to the inner surface of the thick pipe, the value of c = db
, at the inner surface
of the thick pipe, would be around 108. Thus the proximity force approximation will
certainly not be an adequate approximation for the Casimir energy densities near the
inner surface of the thick pipe, and it is necessary to consider the effects of going beyond
the proximity force approximation.
2.4.1 Beyond the proximity force approximation
I shall now consider the effects of including, in the expansions (132), of the c(i) near
the inner surface of the thick pipe, and the expansions (137), of the c̃(i) near the outer
surface of the thick pipe, terms depending on c = ḃ, b̈, and higher derivatives of b, with
respect to y, in the case of (132), and terms depending on ȧ, ä, and higher derivatives
of a, with respect to y, in the case of (137). For definiteness, I shall consider (132),
near the inner surface of the thick pipe, with similar considerations applying to (137),
near the outer surface of the thick pipe.
Now the terms proportional to b−8, in (132), get contributions (159), from the
classical energy-momentum tensor, (158), of the three-form gauge field, CIJK , for the
case of smooth compact quotients of CH3, and contributions from the t8t8R
4 term,
in the low energy expansion of the quantum effective action of supergravity in eleven
dimensions. The proximity force approximation is in fact exact, for the three-form
gauge field configuration (150), but we expect there to be contributions involving c, b̈,
, and d
, coming from the metric variation of the t8t8R
4 term.
As a guide to the derivatives of b with respect to y that might be expected, and the
powers to which they might occur, at higher orders in the expansion in κ
3 , in (132),
we note that for even n ≥ 0, the term C(i)n κ
(n−1)
b8+3n
, in (132), could come from terms
built from
4 + 3
Riemann tensors, in the low energy expansion of the quantum
effective action. Considering, first, just the powers of c that might occur, we see, from
the Riemann tensor components, (96), that each power of 1
, can bring in up to one
power of c. If we extend this to odd n ≥ 1 as well, and bear in mind that for the bulk
power-law solution, (206), c will be very large compared to 1, near the inner surface
of the thick pipe, the strongest dependence on c, that we expect at order κ
(n−1), is
(n−1) c8+3n
b8+3n
We now need to determine the range of values of b, and of y, where such a term could
significantly alter the results of the study of the Einstein equations, and the boundary
conditions for the metric, in the preceding subsections. If we consider the second Ein-
stein equation, in the form (203), the ratio of c
, to the new term, will be
)6+3n
And for b small compared to B, in (206), we have c ≃
)1.8990
, almost right up to the
inner surface of the thick pipe, according to subsection 2.4.3. Thus the ratio of c
the new term, will be
)2.8990 (
)1.8990
)6+3n
)0.6551
)2.8990×(6+3n)
This is larger than 1, for
)0.6551
. And, for B
≫ 1, this will be for most
of the range κ2/9 < b < B. And by (207), ignoring factors of order 1, y > κ2/9 implies
b > B
)0.3449
= κ2/9
)0.6551
. Thus the new terms, involving c, will be signifi-
cant for y < κ2/9, and will be likely to alter the conclusions of subsection 2.4.3, about
this region, but they will be negligible for y ≫ κ2/9, which for B
≫ 1, will be most of
the bulk. We note that the point where b
)0.6551
, and y ∼ κ2/9, is the point
where b
Considering, now, terms involving higher derivatives of b, with respect to y, we see,
from (96), that in addition to terms proportional to c
, one Riemann tensor can also
bring in terms proportional to b̈
, which, by (205), is ∼ c2
in the first bulk power
law region, to the extent that (204), and (205), are not significantly altered by the new
terms. In general, from terms in the low energy expansion of the effective action, built
from polynomials in the Riemann tensor and its covariant derivatives, we expect terms
involving products of expressions c
, . . ., and non-negative powers of
. But by repeated use of (205), we find that 1
, where ∼ means up to constant
factors of order 1. Thus, at each mass dimension (8 + 3n), the largest terms, in the
first bulk power law region, where (205) and (206) are approximately valid, that we
can build by use of factors involving higher derivatives of b with respect to y, are no
larger than the terms κ
(n−1) c8+3n
b8+3n
, whose effect has already been considered.
Thus the effect of going beyond the proximity force approximation, is that the bulk
power law solutions (206), (207), and (210), are no longer expected to be approximately
valid throughout the whole range from b somewhat larger than κ
9 , to b somewhat
smaller than B, but rather, only over the slightly smaller range, from where y ∼ κ2/9,
and b ∼ κ2/9
)0.6551
, to b somewhat smaller than B.
Now as I mentioned just before the start of this subsection, we will find, in sub-
sections 2.6.1 and 2.7, that for TeV-scale gravity, we require B
∼ 1.5×104|χ(M6)|0.1715 . And
from subsection 2.3.6, on page 66, the minimum value of b1 allowed by the Giudice-
Rattazzi-Wells estimate of the effective expansion parameter in quantum gravity in
eleven dimensions [11] is b1 ≃ 0.2κ2/9, which means that |χ (M6)| cannot be larger
than around 7× 104. Thus B
∼ 104, so if the bulk power law (206) continued to be
valid until very close to the inner surface of the thick pipe, we would find c = db
∼ 108
near the inner surface of the thick pipe.
Thus it is clear that the proximity force approximation, in which the Casimir co-
efficients in (132) are approximated by their values as calculated on flat R5 times the
compact six-manifold, will not, in fact, be an adequate approximation. One way to
take account of this would be to generalize the expansions (132), so as to include ex-
plicit dependence also on c = db
, and on higher derivatives of b with respect to y, as
in the order of magnitude estimates above. However this is not an appropriate way
to study the detailed form of the quantum corrections, just as it is not appropriate to
study the relativistic corrections to the Schrödinger equation for atoms, by expanding
m2 + ~p2 ≃ m + ~p2
+ . . ., with ~p interpreted as −i~∂, to higher orders in ~p, because
this results in differential equations of higher and higher order, and correspondingly,
larger and larger numbers of constants of integration, making it difficult to single out
the particular solution of physical interest.
Instead, the appropriate way to study the quantum corrections is to use an iterative
approach, calculating the Casimir corrections for a trial form of b (y) in the quantum
region b1 ∼ κ2/9 ≤ b ≤ κ2/9
)0.6551
, and expressing the results, in the approximation
of neglecting dependence on position in the compact six-manifold, or in other words, of
neglecting all but the leading term, in the harmonic expansions of Lukas, Ovrut, and
Waldram [67], as expansions of the form (132), depending only on b, and not on c, or
any higher derivatives of b with respect to y, but with coefficients that now depend on
the trial form of b (y) in the quantum region, and possibly, also, intermediate powers
of b, not present in (132), then solving the field equations and boundary conditions
with these Casimir coefficients, and if the resulting b (y) differs from the trial b (y),
repeating the process with a new trial b (y), until a self-consistent solution is found for
b (y) in the quantum region, that joins smoothly onto the bulk power law (206), with
the required value of B
∼ 104 for TeV-scale gravity, at b≫ κ 29
)0.6551
. Of course,
the possibility of finding such a self-consistent b (y), in the quantum region, is likely to
depend on the choice of the compact six-manifold M6.
2.4.2 The region near the inner surface of the thick pipe
We now consider the region near the inner surface of the thick pipe, to find out whether
a value of B greater than around 105κ2/9 could occur, as required for TeV-scale gravity.
From the discussion above, we know that the proximity force approximation will not be
adequate. However, we can start by assuming that the t(i) functions have expansions
of the form (132), on page 61, near the inner surface, and see whether the solution of
the Einstein equations can self-consistently reproduce the t(i) functions that we started
with, and also produce the required large value of B
. I shall consider first the case
where t(3) = t(1), near the inner surface of the thick pipe, so the C(2)n and C
n will be
given by (134), in terms of the C(1)n .
We recall from subsection 2.3.2, on page 35, that the value b1 of b at the inner
surface of the thick pipe cannot be larger than around 1.2κ2/9, which corresponds to
|χ (M6)| ≃ 1, and from subsection 2.3.6, on page 66, that it cannot be smaller than
around 0.2κ2/9, which corresponds to |χ (M6)| ≃ 7× 104.
Let us consider, first, the case where all the C(1)n are zero, except for a single value
of n. Then (203) becomes:
(n+2)
b8+3n
4 + n
C(1)n
(n+2)
b8+3n
= 0 (211)
for some fixed value of n ≥ 0. Let us now consider b in the region
∣C(1)n
κ2/9 (212)
so that we can neglect the term − 8
in the square root, and the term 4
, in (211).
Then (211) becomes:
(n+2)
b8+3n
4 + n
C(1)n
(n+2)
b8+3n
= 0 (213)
Let us try for a power law trajectory,
c = σ
, (214)
for some numerical constant σ, and exponent ρ. We then find that C(1)n must be
negative, which is satisfied for the energy-momentum tensor of the three-form gauge
field, (159), and corresponds to a positive contribution to the energy density, T00, and
that:
ρ = −
(6 + 3n)
, σ =
n (4 + n)
3 (16 + 24n+ 3n2)
(215)
Thus we see that, in contrast to the situation for the bulk-type trajectories, where
every trajectory is approximately a power law trajectory, for a certain range of b, that
depends on the trajectory, there is now just a single power law trajectory. If we now
try for a solution of the form c = c0 (1 + s), where c0 is the power law trajectory, and
s (b) is a small perturbation, we find that
s = Sbη, η =
16 + 24n+ 3n2
2 (4 + n)
(216)
where S is a constant of integration. Now η ≥ 2 for n ≥ 0, so as we follow trajectories
near the power law trajectory, in the direction of decreasing b, they tend to converge
towards the power law trajectory, in the sense that s decreases in magnitude, so in this
sense, the power law trajectory is an attractor, in the direction of decreasing b.
Now in the proximity force case, using t(3) = t(1) and (133), the equation (203) can
be written, in the region where −κ2t(1) ≫ 1
, as:
κ2t(1) +
−2t(1) + b
dt(1)
= 0 (217)
The small b power law solutions, (214) and (215), can all be written as:
c2 = −b
κ2t(1) (b)− 4
κ2t(1) (x) (218)
The integral is convergent at x → ∞, because t(1) (x) decreases at least as rapidly as
x−8, as x → ∞. Now (218) does not give an exact solution of (217), except when
t(1) (b) is a pure power of b. In fact, on substituting (218) into (217), the left-hand side
of (217) reduces to:
κ2t(1) +
 κ2t(1) (x) (219)
When t(1) (b) is not a pure power of b, there are cross terms between different powers
of b, that do not cancel out of the first term in (219), but the second term in (219) is a
linear combination of the contributions from different powers of b. However, if t(1) (b)
is a linear combination of two different pure powers, say b−(8+3n) and b−(8+3m), with n
and m large, then the remainder term, (219), is ∼ 1√
, while the leading terms, in
(217), are ∼ n or m. Furthermore, for a pure power b−(8+3n), with large n, the integral
term, in (218), is of order 1
compared to the leading term.
Thus it seems likely that, when the coefficients in t(1) (b) are all ≤ 0, a reasonable
approximation to the small b attractor trajectory, generalizing the small b power laws
(214), (215), valid when t(1) (b) is a pure power, will, in the limit of large −t(1) (b), be:
κ2t(1) (b) (220)
Now for any sufficiently large value of c, the trajectory passing through the point
(b, c) will still be of the bulk power law type (206), even for b in the range (212).
However, when all the coefficients in t(1) (b) are ≤ 0, any power law trajectory, of the
type (206), will intersect the small b attractor trajectory, (220), for sufficiently small b.
From the perturbative analysis carried out in connection with (216), it is clear that what
actually happens, when all the coefficients in t(1) (b) are ≤ 0, is that each bulk power
law trajectory, (206), curves upwards as it approaches the small b attractor trajectory,
(220), and then approaches the small b attractor trajectory gradually, without ever
actually crossing it.
Now regarding a as a function of b again, and considering the case where C(1)n is
only nonzero, for the same n as in (211), the equation (194) for ȧ
, with the upper
choice of sign, and dropping the term 4
in the square root, becomes:
= −2c
(n+2)
b8+3n
(221)
On the unique small b power law trajectory, defined by (214) and (215), we can neglect
the − 8
term in the square root in this equation, which then becomes:
(4 + 2n)
(4 + n)
(222)
Hence:
a = A1
(223)
where A1 is another constant of integration. For n = 0, the solution defined by (214),
(215), and (223), which corresponds to b = βκ2/9
, a = A1
, where β is
a constant, has the functional form of the supersymmetric solution found by Lukas,
Ovrut, Stelle, and Waldram [68], for the case when the compact six-manifold is a
Calabi-Yau threefold with h11 = 1, transformed to the coordinate system where the
metric has the form (94).
We now have to consider whether these solutions can be self-consistent, when we
recalculate the expansion coefficients C(i)n in (132) for b (y) corresponding to these solu-
tions, in accordance with the discussion in the preceding subsection. We can no longer
assume that t(3) = t(1), but since we are now just considering orders of magnitude,
it will be adequate to consider the case where t(3) = t(1). Let us suppose that in the
quantum region, where b
)0.6551
, we have a power law, c =
, with γ ≥ 0,
which joins continuously onto the bulk power law (206), at b
)0.6551
. Then
)0.6551(1+γ)
≃ 4001+γ, (224)
where I used that B
≃ 104 for TeV-scale gravity. Now by the preceding subsection,
we expect a term κ−
9 C(i)n
)8+3n
, in (132), to be accompanied by an additional
term ∼ κ− 229 C(i)n
)8+3n
c8+3n. This now becomes:
9 C(i)n
)8+3n
)γ(8+3n)
9 C(i)n
)(1+γ)(8+3n)
(225)
and thus contributes expansion coefficients C
ñ ≃ 4008+3ñC(i)n to the recalculated t(i),
where ñ = (1 + γ)n + 8
γ. If we now consider the case where C
ñ is significant only
for one value of ñ, and assume that the significant C
ñ is negative, then by (214) and
(215), the recalculated c, calculated from the recalculated t(i), is
∼ 4004+
)3+ 3
, (226)
where I dropped all factors of order 1. This is in agreement with the c we started with
at the upper limit of the quantum region, where b
≃ 400, but for all γ ≥ 0, and all
n ≥ 0, increases much more rapidly with decreasing b than the c we started with, and
for b ∼ κ2/9, is very large compared to the c we started with. We would not expect
the discrepancy to be any smaller if more than one C
ñ is significant, provided all the
significant C
ñ are negative. Thus we cannot obtain a self-consistent solution if all the
significant C
ñ are negative.
Now since the proximity force approximation is not valid in the quantum region
b1 ∼ κ2/9 ≤ b ≤ κ2/9
)0.6551
, we cannot assume that t(3) = t(1), but for the purpose
of illustration, I shall continue to consider the case where t(3) = t(1). Then by the
result above, if a self-consistent solution with B
≃ 104 exists, the self-consistent t(1),
in (130) and (132), must contain at least one significant C
ñ that is positive, which
corresponds to a negative contribution to the energy density T00. This is expected
to be possible for Casimir energy densities, whose sign often depends on the detailed
geometry of a physical situation [258], although recent results of Kenneth and Klich
[259] and Bachas [260] have shown that Casimir forces are always attractive in certain
circumstances.
If t(1) is dominated by a single term κ−
9 C(1)n
)8+3n
in (132), where C(1)n is
positive, then as noted after (214), there is no small b power law solution of (213), for
that C(1)n . Instead, the generic solution of (213), with C
n > 0, with c viewed as a
function of b, in the quadrant b > 0, c > 0 of the (b, c) plane, has a peak at a point
(bp, cp) where
6n2 + 36n+ 64 + 3n + 4
)3+ 3n
. (227)
Every point in the quadrant b > 0, c > 0 of the (b, c) plane must now lie on a
trajectory that has such a peak, for if dc
is positive at the point (b, c), and we follow
the trajectory in the direction of increasing b, the terms in (213) proportional to 1
b8+3n
will eventually become negligible, and the trajectory will then take the form (206),
for some B > 0, so that dc
is now negative. Now suppose that dc
is negative at the
point (b, c), and follow the trajectory in the direction of decreasing b. If C(1)n
(3n+6)
b8+3n
small compared to c
, then the trajectory has the form (206), so C(1)n
(3n+6)
b8+3n
increases
more rapidly than c
with decreasing b, and a value of b > 0 will be reached where
the two terms are comparable in magnitude. Then either the two terms continue
to be comparable in magnitude as b decreases further, or C(1)n
(3n+6)
b8+3n
becomes large
compared to c
, as b decreases further. But if the two terms continue to be comparable
in magnitude as b decreases further, then we have c
≃ αC(1)n κ
(3n+6)
b8+3n
, for some constant
α > 0, for all b from the value > 0 where the two terms first become comparable in
magnitude, down to b = 0. But this is the characteristic property of the small b power
law trajectory (214), (215), and the trajectories that asymptotically approach it, in the
direction of decreasing b, in the sense described after (216), and, as noted after (214),
there is no small b power law trajectory for C(1)n > 0. Thus C
(3n+6)
b8+3n
must become
large compared to c
, as b decreases further, beyond the value > 0 where the two terms
are comparable. The trajectory then tends to the form
(4 + n)
9 (2 + n)
)6+3n
)6+3n
, (228)
where bs > 0 is a constant of integration, so that
is positive.
Now if such a peak occurs, then for self-consistency, when we include the Casimir
energy density corrections beyond the proximity force approximation, as discussed in
the preceding subsection, the peak must occur at the upper limit of the quantum re-
gion, so bp ∼ 400κ2/9. This is because we have the bulk power law (206) to the right of
the peak, and from the discussion above, we cannot self-consistently have any power
law c ≃
, with γ ≥ 0, in the quantum region. The peak will be broad, with width
∼ bp, so in the region of the peak, we can treat c as a constant ∼ 400. The additional
term ∼ κ− 229 C(1)n
)8+3n
c8+3n, which by the preceding subsection, we expect to ac-
company the term κ−
9 C(1)n
)8+3n
in (132), now becomes ∼ κ− 229 C(1)n
)8+3n
Substituting this into the right-hand side of (227), and dropping all factors of order 1,
we see that we have self-consistency in the region of the peak. However, from compar-
ison of (227) and (228), we see that
cannot be large compared to 1, because if
was much larger than 1, (228) would allow c to become substantially larger than the
maximum value given by (227), in the region where (228) is still valid. Thus since b1
cannot be smaller than bs, we cannot obtain a self-consistent result with b1 ∼ κ2/9, and
bp ∼ 400κ2/9, in this way.
A similar result is also expected when no single term is dominant in t(1) in (132),
because if the C(1)n term in t
(1) is multiplied by c8+3n for all n ≥ 0, with c a constant > 1,
the effect is to multiply the minimum possible value of b, as derived in subsection 2.3.6,
on page 66, from the Giudice-Rattazzi-Wells estimate [11] of the effective expansion
parameter for quantum gravity in eleven dimensions, by c.
Since there is no difficulty obtaining self-consistency at the upper limit b ∼ 400κ2/9
of the quantum region, but we cannot obtain consistency inside the quantum region
for any power law c ≃
, with γ ≥ 0, we now try for a power law of this form
with γ < 0. We see that if γ = −1, corresponding to a linear dependence of c on b,
then ñ = (1 + γ)n + 8
γ is independent of n, and equal to −8
. We can now simply
have Bq ≃ κ2/9, in which case, if the magnitude of C(1)n in (132) is ∼ 0.23nC(1), for
some constant C(1) of order 1, as suggested by the minimum value of b estimated in
subsection 2.3.6, the additional terms ∼ κ− 229 C(1)n
)8+3n
c8+3n sum up to no more
than around κ−
9 (1− 0.23)−1C(1) ≃ κ− 229 C(1). And considering the equation (213) for
n = −8
, we see from (214) and (215) that we do indeed have a unique linear solution,
, (229)
provided that the effective C
is found to be positive. We see that C
will be
self-consistently determined as a fixed number of order 1, provided that this number is
positive. Thus it seems reasonable to expect that for around fifty percent of all possible
choices of a smooth compact quotient M6 of CH3 or H6 that is a spin manifold, a
spin structure on M6, and a topologically stabilized configuration of vacuum Yang-
Mills fields on the inner surface of the thick pipe, consistent with Witten’s topological
constraint [45], a value of B larger than κ
9 will be found by this mechanism.
The actual value of b at which the self-consistent quantum linear relation (229)
transforms into the classical relation (206), and the corresponding value of B, will be
determined by how close to the self-consistent quantum linear relation (229) the system
is set by the boundary conditions at b = b1 ∼ κ2/9. We note that η, in (216), is equal to
−10 when n = −8
, so the linear solution (229), of (213) with n = −8
, is a very strong
attractor in the direction of increasing b. However this has not taken into account the
fact that in the presence of deviations from the self-consistent linear relation (229), the
equation to be solved will no longer be precisely (213), with n = −8
. We also note,
from the discussion above, that it is consistent for c to be approximately constant in
the region of the peak at b ∼ κ2/9
)0.6551
, although not for b≪ κ2/9
)0.6551
we expect the transition from (229) to (206) to occur smoothly across a broad peak of
width ∼ κ2/9
)0.6551
The linear relation (229) means that b depends exponentially on y in the quantum
region:
b = bq exp
(y − yq)
, (230)
where bq ≡ κ2/9
)0.6551
and yq ≡ 0.3449κ2/9, for agreement with (207) at b = bq, for
y0 = 0. The thickness in y of the quantum region is ∼ κ2/9 ln bqκ2/9 , which for TeV-scale
gravity, with bq greater than around 2000κ
9 , is ∼ 8κ2/9.
We note that if the percentage of possible choices of M6, its spin structure, and
the vacuum Yang-Mills fields, for which the thickness in y of the quantum region is
greater than a certain value, decreases roughly exponentially with that value, then the
percentage of possible choices, for which B
is greater than a certain value, will be
roughly given by a fixed negative power of that value.
From (222) and (223), with n = −8
, we see that in the quantum region, where the
linear relation (229) applies, a also depends linearly on b:
a = A1
, (231)
where A1 is a constant of integration. However this linear dependence of a on b in the
quantum region, for n = −8
, is a consequence of the proximity force relation t(3) = t(1),
which would apply for compactification on flat R5 times the compact six-manifold M6,
and as noted above, there is no reason to expect this relation to hold when a and b
depend nontrivially on y. Consideration of the special case where this relation holds
was adequate for the order of magnitude studies above, where only the dependence
of b on y was considered, but to determine the possible dependences of a on b in the
quantum region, I shall now assume that the t(i), in (130), on page 58, are constrained
only by the conservation equation (131).
Considering the region κ2/9 ≪ b≪ bq, only the terms κ−
, in the self-consistent
versions of the expansions (132), on page 61, will be significant. The relevant equations
are now (194) and (195), on page 87, with the upper choice of sign, and
t(i) ≃ κ−
, (232)
where the C
are numerical constants, to be determined self-consistently, as discussed
above. The only possible power-law dependence of c on b, with this form of the t(i), is
again c = σ b
, where σ is a numerical constant, and this linear dependence of c on b
leads self-consistently to the form (232) of the t(i) in this region, as before. However a
no longer has to depend linearly on b in this region, so we try an ansatz
a = A1
. (233)
The conservation equation (131) then reduces to:
(4τ + 6)C
− 4τC(1)− 8
− 6C(2)− 8
= 0. (234)
Choosing C
and C
as independent, equations (194) and (195) reduce in this region
(2τ + 4)σ =
6σ2 +
(235)
− σ2 + σ
6σ2 +
− (τ + 2)C(1)− 8
+ (τ + 1)C
= 0, (236)
from which we find:
(2τ + 4)
2 − 6
(237)
(2τ + 3) σ2 =
(τ + 2)C
− (τ + 1)C(3)− 8
. (238)
Thus almost any value of τ can be obtained, if there exists a suitable smooth compact
quotient M6 of CH3 or H6 that is a spin manifold, and a choice of a spin structure on
M6 and a topologically stabilized configuration of the Yang-Mills gauge fields on the
inner surface of the thick pipe, that results self-consistently in the appropriate values
and C
. In particular, the bulk power law value τ = −0.7753 is one of the
two solutions if C
> 0 and C
= 0. However τ = −2 would imply that the square
root vanished, so that we could not conclude that all three Einstein equations would
be satisfied.
The calculation of C
and C
for a particular example requires, in particular,
the calculation of the propagators and heat kernels for all the CJS fields on a flat R5
times uncompactified CH3 or H6 background, as appropriate. These can be obtained
from the corresponding propagators and heat kernels on a flat R5 times CP3 or S6
background, which can be calculated by using the Salam-Strathdee harmonic expan-
sion method [244], and summing the expansions by means of a generating function.
This calculation is currently in progress for CH3, and the scalar heat kernel on CH3,
obtained by this method, is presented in subsection 4.1, on page 196.
For t(3) 6= t(1), we can no longer study trajectories near the self-consistent linear
trajectory c = σ b
by perturbing only the dependence of c on b as c = σ b
(1 + Sbη),
where S is a small constant of integration. The dependence (233) of a on b also has to
be perturbed as a = A1
(1 + Ubη), where U is a small constant of integration,
and the t(i) functions (232) in the region κ2/9 ≪ b ≪ bq have to be perturbed as
t(i) ≃ κ− 229 C(i)− 8
1 + V (i)bη
, where the V (i) are small constants. The Einstein equations
(194) and (195) and the conservation equation (131) impose three relations among the
six constants describing the perturbation, and we would now expect the exponent η
to depend on ratios of the small constants S, U , and the V (i), rather than having the
unique value −10 as for the case when t(3) = t(1).
The possibility of having both a self-consistent quantum region, in which c increases
linearly with b as σ b
, with σ a numerical coefficient of order 1, and a self-consistent
classical region where c satisfies the classical bulk power law (206), on page 89, is due
to the presence, beyond the proximity force approximation, of additional terms
9 C(i)n,n
κ2/9c
)8+3n
, (239)
with n ≥ 0, in the expansions (132), on page 61, of the t(i) functions in (130), on page 58.
These terms sum to finite constant terms κ−
at low orders of perturbation theory
in the quantum region, provided σ is not too large, and thus result self-consistently in
the linear dependence of c on b in the quantum region, provided the C
are consistent
with σ2 > 0, as determined by (237) and (238). While if c is related to b by the classical
bulk power law (206), and b is larger than bq = κ
)0.6551
, so the value of c given
by (206) is smaller than the value that would be given by extrapolating the linear
relation from the quantum region, then the terms (239) rapidly decrease in magnitude
with further increase in b, and quickly become negligible, so that the classical bulk
power law (206) becomes self-consistent.
Thus it is consistent for the quantum region to transform into the classical region at
any point bq > b1, and the value of bq at which the transition occurs in a particular ex-
ample, and consequently the value of B, will depend on how close to the self-consistent
linear trajectory c = σ b
, with σ determined by (237) and (238), the system is set by
the boundary conditions at b = b1, and on whether the self-consistent linear trajectory
attracts or repels neighbouring trajectories, in the direction of increasing b, and how
strongly it does so. From the discussion above, we see that for t(3) 6= t(1), the space of
relevant neighbouring trajectories is three-dimensional, and parametrized, for example,
by small quantities S, V (1), and V (3). The actual transition from the quantum region
to the classical region will take place gradually, over a broad peak of width around bq,
as discussed just before (230).
The presence of the additional terms (239) in the t(i) functions, beyond the prox-
imity force approximation, follows from their presence in the local terms formed from
powers of the Riemann tensor, and the components (96), on page 34, of the Rie-
mann tensor for the metric ansatz (94). In particular, RABC
D contains both a term
D (h), which for a local term in the quantum effective action Γ formed from 4+3m
powers of the Riemann tensor leads both for CH3, on using the CHn Riemann tensor
components (71), on page 28, and also for H6, to terms in the t(i) functions of the
form κ−
)8+6m
, in agreement with the even order terms in (132), and a term
GACδB
D, which for the same term in Γ leads to even order terms of the form (239).
We note, furthermore, that since, on a power law trajectory, ȧ
is equal to ḃ
times
a fixed number of order 1, the RµAν
B and RAµB
ν components, and the ȧ
terms in
τ , will lead both in the quantum region and the classical region to terms similar in
magnitude to the terms (239). And since ä
and b̈
are equal, on a power law trajectory,
to ḃ
times fixed numbers of order 1, except that b̈
vanishes on the self-consistent linear
trajectory in the quantum region, the Rµyν
y and Ryµy
ν components will also lead both
in the quantum region and the classical region to terms similar in magnitude to the
terms (239), and the RAyB
y and RyAy
B components will lead in the classical region to
terms similar in magnitude to the terms (239).
Now by definition, the metric gµν , in the metric ansatz (94), has de Sitter radius
equal to 1. Hence the value a1 of a, at the inner surface of the thick pipe, is equal
to the observed de Sitter radius (22). We recall that in subsection 2.3.2, on page
35, we found, by combining an estimate of the d = 4 Yang-Mills coupling constants
at unification, with the Hořava-Witten relation (45), that b1
≃ 1.2772
|χ(M6)|
, when the
compact six-manifold M6 is a smooth compact quotient of CH3, and b1
≃ 1.1809
|χ(M6)|
when M6 is a smooth compact quotient of H6. Thus from (233), we find:
|χ (M6)|
1.2772
× de Sitter radius, (240)
when M6 is a smooth compact quotient of CH3, and the same relation, with 1.2772
replaced by 1.1809, when M6 is a smooth compact quotient of H6.
And from matching the bulk power law (210) for a in terms of b to (233), at b = bq,
we find:
A = A1
)τ+0.7753
)0.6551τ+0.5079
. (241)
Thus:
|χ (M6)|
1.2772
)0.6551τ+0.5079
× de Sitter radius (242)
for a smooth compact quotient of CH3, and the same relation, with 1.2772 replaced
by 1.1809, holds for a smooth compact quotient of H6.
2.4.3 The boundary conditions at the inner surface of the thick pipe
Now, treating b as the independent variable, the boundary conditions (169), on page
82, become:
b=b1+
−5t̃[1](1) + 6t̃[1](2)
b=b1+
4t̃[1](1) − 3t̃[1](2)
(243)
where b1 ≡ b (y1). The coefficients t[1](i) receive contributions from the Lovelock-Gauss-
Bonnet terms, given by (188), on page 85, for quotients of CH3, and by (192) for
quotients of H6; from the leading terms in the Lukas-Ovrut-Waldram harmonic expan-
sion, on the compact six-manifold M6, of the energy-momentum tensor of topologically
stabilized vacuum Yang-Mills fields on the inner surface of the thick pipe; and from
Casimir effects on the inner surface of the thick pipe.
The terms in (188) and (192) that involve negative powers of a are negligible at the
inner surface of the thick pipe, so the Lovelock-Gauss-Bonnet terms, for quotients of
CH3, are:
t̃[1](1)LGB =
λ2b41
, t̃[1](2)LGB =
λ2b41
(244)
and for quotients of H6, they are:
t̃[1](1)LGB =
λ2b41
, t̃[1](2)LGB =
λ2b41
(245)
It would seem reasonable to expect that the contributions to the coefficients t̃[1](i),
from the leading terms in the Lukas-Ovrut-Waldram harmonic expansion of the energy-
momentum tensor of topologically stabilized vacuum Yang-Mills fields on the inner
surface of the thick pipe, will be roughly a positive numerical multiple of the contribu-
tions that result from embedding the spin connection in the gauge group for CH3, as
given in (181), on page 84, for the outer surface of the thick pipe. Thus we estimate
the vacuum Yang-Mills field contribution to the coefficients t̃[1](i) as:
t̃[1](1)YM ≃ − 24
λ2b41
N, t̃[1](2)YM ≃ − 8
λ2b41
N, (246)
where the numerical constant N ≥ 0 is given by
96V (M6)
hhCDhEF trF
DF , (247)
in terms of the topologically stabilized vacuum Yang-Mills fields on the inner surface of
the thick pipe, with V (M6) given by (99), on page 36, for a smooth compact quotient
of CH3, and by (100), for a smooth compact quotient of H6.
Now the results (188), (192), (244), and (245), for the Lovelock-Gauss-Bonnet con-
tributions, have been calculated assuming that the Riemann tensor in the Lovelock-
Gauss-Bonnet term in (48), on page 23, is the d = 10 Riemann tensor calculated from
the induced metric GUV on the Hořava-Witten orbifold hyperplanes, and not the re-
striction to the orbifold hyperplanes of the d = 11 Riemann tensor. This would seem to
be a reasonable assumption, because it implies that for a Calabi-Yau compactification
[9], with the standard embedding of the spin connection in the gauge group, the Rie-
mann tensor term in (48) is automatically equal to −1
times the Yang-Mills term, at
an arbitrary point of the Calabi-Yau moduli space. For the compactifications consid-
ered here, it implies that when we go beyond the proximity force approximation, there
are no related terms with factors of c = db
that can cancel the 1
factor in the region
κ2/9 < b < bq = κ
)0.6551
, where, from the previous subsection, c is ∼ b
If we assume, by analogy with this, that when we go beyond the proximity force
approximation, there are also no terms related to the higher order terms in the expan-
sions (136), on page 62, with enough powers of c to cancel all the powers of 1
in those
terms in the region κ2/9 < b < bq, then the boundary conditions (243), at the inner
surface of the thick pipe, cannot be solved for any value of b1 much larger than κ
This is in agreement with the result from subsection 2.3.2, on page 35, that to fit a
reasonable estimate of the unification value of the observed d = 4 Yang-Mills coupling
constants, the value of b1 cannot be larger than around 1.2κ
2/9, which corresponds to
|χ (M6)| ≃ 1.
On the other hand, it would seem reasonable to expect that for perhaps around
three percent of choices of a smooth compact quotient M6 of CH3 or H6 that is a
spin manifold, a spin structure on M6, and a topologically stabilized configuration
of vacuum Yang-Mills fields tangential to M6, a solution of the boundary conditions
(243) will exist with B larger than around 5κ2/9, and b1 no smaller than around twice
the minimum value ≃ 0.2κ2/9 derived in subsection 2.3.6, on page 66, from the GRW
estimate [11] of the expansion parameter of quantum gravity in eleven dimensions, so
that the boundary conditions can be solved perturbatively.
For let us suppose that we have done the one-loop calculation for a trial classical
metric GIJ , and have an approximation to the expansions (132), on page 61, of the t
that contains terms with at least two different powers of b. Then from the preceding
subsection, we expect there to be roughly a fifty percent chance of having at least a
small region b1 < b < bq in which c increases roughly linearly with b, so that the lowest
power of 1
in the self-consistent t(i) will be zero, as in (232). The approximation to the
t(i) contains only a few terms, so for b somewhat smaller than κ2/9, it will be dominated
by the terms with the largest power of 1
, and there is around a fifty percent chance
that these will lead to a small b power law trajectory.
And similarly, the perturbative approximation to the expansions (136), on page 62,
will contain only a few terms, so in this approximation, the ratio of the right-hand
sides of the boundary conditions (243) will have an approximately fixed value for b
somewhat larger than κ2/9, and generically some other approximately fixed value for b
somewhat smaller than κ2/9.
Now on any power law trajectory in the bulk, da
is a fixed multiple of a
, where the
fixed multiple is characteristic of the trajectory, so c
is a fixed multiple of c
on the
trajectory. Thus in this approximation the boundary conditions (136) generically have
no simultaneous solution for any value of b1 either much larger or much smaller than
κ2/9, while there is perhaps a fifty percent chance there will be a solution in the region
with b1 ∼ κ2/9 where each term is ∼ 1 in magnitude, and the ratios of the left-hand
sides and right-hand sides are moving between their limiting values.
Finally, if there is such a solution of the boundary conditions in this approximation,
we would expect there to be roughly a fifty percent chance that it will have b1 greater
than the minimum value of around 0.2κ2/9 estimated in subsection 2.3.6, on page 66,
and perhaps another fifty percent chance that it will have b1 greater than around twice
this value, so that the one-loop calculation would give a reasonable approximation to
the correct result.
2.4.4 The classical solutions in the bulk
I shall now consider solutions of the Einstein equations in the classical part b > bq =
)0.6551
of the bulk, that start out in the classical region on a trajectory of
the form (206), on page 89, with B large compared to κ2/9, and follow such solutions
further into the bulk, towards the region where c is no longer large compared to
so that (204) no longer reduces to (205). From (209) and (210), we know that a is
decreasing in magnitude, as b increases, in this region, and, depending on the values of
the integration constant B, in (206), which is determined by the boundary conditions
at the inner surface of the thick pipe, and the integration constant A, in (210), which
is not determined by the boundary conditions at the inner surface of the thick pipe,
and is at present a free parameter, that will eventually be determined by the boundary
conditions at the outer surface of the thick pipe, we may or may not have to stop
neglecting the term 4
, in the square root (197), as it occurs in the classical Einstein
equation (204), before we reach the region where c is no longer large compared to
I shall first consider the case where the term 4
, in the square root (197), continues
to be negligible, into the region where c is no longer large compared to
, so the
equation to study is (204).
We first note that (204) has the solution c =
. However, when the terms 4
and 2
κ2t(3) are negligible in the square root R, defined in (197), as presently assumed,
R vanishes identically for this solution, hence we cannot conclude, from (198), that
(194), (195), and (131) imply that all three Einstein equations are satisfied. And
indeed, this special solution, of (204), does not correspond to a solution of all three
Einstein equations.
Now the second term in the parentheses, in (204), is smaller in magnitude that
times the first term in the parentheses, for all c ≥
, and tends to 0 relative to the
first term as c→
from above. Hence for c ≥
, but near
, (204) reduces to:
2 (3c2 − 4) (248)
The solution of this is:
 (249)
where B1 > 0 is a constant of integration, different from the constant of integration,
B, in (206). Now (249) gives:
 (250)
This is negative for b < B1, vanishes for b = B1, at which point c =
, and positive for
b > B1. Thus we see that the solutions (249), with different values of B1, all osculate
with the line c =
, at different points along this line, and that, moreover, as each
solution (249) passes the point b = B1, in the direction of increasing b, it moves from
the positive sign to the negative sign of the square root, in (248), or in other words,
from the upper sign, to the lower sign, of the square root, in (199). dc
now becomes
positive, so, if the term 4
, in the square root, R, defined in (197), remains negligible,
c now starts increasing without limit, and, when c is large compared to
, we reach
another power law region, where, instead of (205), (206), and (207) we have (202), and
)7.8990
(251)
for some B2, which in general will be different from both B and B1. (251) corresponds
6 + 2
y3 − y
≃ 0.7558
y3 − y
)0.1449
(252)
for some y3 > y1.
Now we already found, in the first region of the classical part of the bulk, where we
choose the upper sign in (194), (195), and (199), and c is sufficiently large, that we are
on a bulk power law trajectory (205), (206), and (207), that a depends on b through
a power law, (209) and (210), so that a decreases as b increases. In the second bulk
power law region, where we have the lower sign in (194), (195), and (199), and c is
sufficiently large, that we are on a bulk power law trajectory (202), (251), and (252),
we have:
= −3.2247
(253)
so that:
a = A2
)3.2247
(254)
where A2 is constant, which will in general be different from A, in (210), so that a now
decreases much more rapidly, with increasing b, than it did in the first bulk power law
region, (209) and (210).
A convenient interpolating function, that agrees with (207) in the first bulk power
law region, when y0 is set to 0, and agrees in form with (252), in the second bulk power
law region, is:
(1− αy)
= 1.4436
)0.3449
(1− αy)0.1449
(255)
To fix α, we note that, from (201) and (204), cdc
= b̈ should vanish, when c = ḃ =
The zero of b̈, at y < 1
, is at:
y = −
6− 387
6− 241
0.4747
(256)
Imposing the requirement that ḃ =
at this value of y, we find that:
0.9094
(257)
Thus in terms of B, the zero of b̈, at y < 1
, is at:
y ≃ 0.4747
= 0.5220B (258)
Thus B1, in (249), which is the value of b, at the zero of b̈, is given by:
B1 ≃ 1.2664B (259)
In the second bulk power law region, the interpolating function, (255), approxi-
mately reduces to:
6 + 2
6 + 1
(260)
Comparing with the bulk power law in the second bulk power law region, (252), we see
that:
= 1.0996B (261)
6 + 1
B = 1.8327B (262)
A convenient interpolating function for the dependence of a on b, that agrees with
(209) in the first bulk power law region, and agrees in form with (253), in the second
bulk power law region, is:
)(2− 12
)0.7753
1 + 4.9154
)2.4495
) (263)
The coefficient of b
6, in the denominator of (263), has been chosen so that da
= −2a
when c =
, so that b = B1, as follows from (221), on neglecting the Casimir energy
term in the square root. Comparing with (254), we see that:
A2 = 0.2034A
)2.4495
(264)
From (194), with the lower choice of sign, together with (251) and (254), we find
that, in the second bulk power law region, ȧ is related to a, by:
− ȧ ≃
)1.1394
(265)
where:
≃ 2.7944
)0.8777
)6.0550
A2 ≃ 0.002109
)0.8777
)1.4556
A (266)
The power law (265) is analogous to (206), on page 89, so by analogy with the
region near the inner surface of the thick pipe studied in subsection 2.4.2, on page 93,
we would expect it to be possible to realize a large value of Ã
by the occurrence of a
quantum region ãq = κ
)0.5326
> a > a2 ∼ κ2/9 adjacent to the outer surface, in
which −ȧ self-consistently grows linearly with a, and −ȧ and a increase exponentially
with the geodesic distance (y2 − y) from the outer surface. The boundary conditions
at the outer surface will determine both the value a2 of a at the outer surface, and
the integration constant Ã, in (265). The integration constant A is then determined in
terms of Ã, B, and κ2/9, by (266), which then determines the relation between a and
b, by (263), and hence the value of b, at the outer surface of the thick pipe. This then
determines y2, or in other words, the value of y, at the outer surface of the thick pipe,
by (255).
We note that for this type of solution, in which a is comparable to κ2/9 at the outer
surface of the thick pipe, the term 4
, in the square root in the Einstein equations (194)
and (195), which is the only term in (194) and (195) that depends on the existence
and sign of the effective cosmological constant in the four observed dimensions, is not
very important, since it is negligible except near the outer surface, where there will be
Casimir terms of comparable magnitude.
Before considering this type of solution in more detail, I shall now look for solutions
such that both a and b are classical, or in other words, large compared to κ2/9, at the
outer surface of the thick pipe. We will see that in contrast to the solutions where a is
comparable to κ2/9 at the outer surface, the 4
term, in the square root in (194) and
(195), is essential for obtaining this type of solution, which thus will exist only when
the effective cosmological constant, in the four observed dimensions, is greater than
zero.
2.5 Solutions with both a and b large compared to κ2/9, at the
outer surface of the thick pipe
I shall now look for solutions of the Einstein equations, and the boundary conditions
(170) at the outer surface of the thick pipe, such that both a and b are large compared to
κ2/9, at the outer surface, assuming that the boundary conditions at the inner surface
have already been solved, such that in the first bulk power law region, we are on a
trajectory (206), (207), and (210), with a large value of B
, but A not yet determined.
We see from (170), (181), (188), (192), the expansions (138), and the relations (45)
and (46), that when both a and b are large compared to κ2/9, at the outer surface of
the thick pipe, the largest terms, in the right-hand sides of the boundary conditions,
(170), are the Yang-Mills terms, (181), forCH3, and the Lovelock-Gauss-Bonnet terms,
(188), for CH3, and (192), for H6, and that these terms are of order 1
, times κ
, or κ
, and thus negligible. Thus we are now looking for solutions such that
, and ḃ
, are zero, at the outer surface of the thick pipe.
From (194), we therefore require that 8
, at the outer surface of the thick pipe,
for this type of solution, or in other words, b =
2a, at the outer surface of the thick
pipe. Furthermore, the term 4
, in the square root, in (194), arose from the Ricci tensor
of four-dimensional de Sitter space, Rµν (g), in (97), and would have been absent, if the
effective cosmological constant, in the four observed dimensions, had been zero, and
would have had the opposite sign, if the effective cosmological constant, in the four
observed dimensions, had been zero. Thus there will be no solutions, such that both
a and b are large compared to κ2/9, at the outer surface of the thick pipe, unless the
effective cosmological constant, in the four observed dimensions, is greater than zero.
We now have to study the coupled equations (194) and (195), when the t(i) are
negligible, but the term 4
, in the square root, is not negligible. We can still express
the two equations as first order differential equations for a, and c = ḃ, as functions of
b, but the two equations are now coupled. Equation (194), with the upper choice of
sign, now becomes:
6c2 − 8 + 4
(267)
And (195), with the upper choice of sign, now becomes:
3c2 − 4
6c2 − 8 + 4 b
(268)
Qualitatively, when the b
term starts to become significant, in the square root in the
right-hand side of (268) the trajectory, in the (b, c) plane, starts to peel off below the
= 0 trajectory. We are looking for a solution where ȧ
, or in other words, c
, and
, or in other words, c
, both tend to zero, at the boundary, while b
tends to
2, at the
boundary, and b tends to a finite nonzero limit. Thus c→ 0 at the boundary, while da
must remain finite. Then 4 b
needs to increase rapidly enough, to compensate for the
decrease in 6c2, so as to keep 6c2 − 8 + 4 b2
> 0, as c tends towards 0. And as c → 0,
will be determined by the − 4
term, which → ∞.
Now the equation dc
= − 4
has the solution c2 = 8 ln
, where b2 is a constant of
integration, that we would like to identify as b (y2), the value of b at the outer surface
of the thick pipe. This solution applies in the region c → 0, b ≃ b2, b ≤ b2, so we can
expand the logarithm, to find:
(269)
If we now define u ≡ b
, the equations (267), and (268) become:
6c2 − 8 + 4u2
(270)
3c2 − 4
6c2 − 8 + 4u2 (271)
And the boundary conditions, at b = b2, become:
2, c = 0 (272)
We now see that there are two different possible behaviours of u near the boundary,
consistent with (269), (270), and (271). Specifically, expanding u in the small quantity
, as u =
1 + α
, we find, from (269), and (270), that
− α = 3−
6 + 2α (273)
which has the solutions:
α = −5
, α = −3 (274)
We note that the first of these is only a solution, for the particular sign of the square
root in (273), while the second is a solution for both signs of the square root, since
the square root vanishes for it. Both the solutions (274) are consistent with (271), and
substituting one of them into (271), fixes the term in c2, that is quadratic in
Then substitution into (270) fixes the quadratic term in u, and so on.
Now (270) and (271) imply that:
6c2 − 8 + 4u2
6c2 − 8 + 4u2
6c2 + u2
(6c2 − 8 + 4u2)
(275)
Hence 6c2 − 8 + 4u2 = 0 is a solution of (270) and (271). However, the square root,
R, defined in (197), vanishes identically for this solution, when the t(i) are negligible,
so we cannot infer, from (198), that 6c2 − 8 + 4u2 = 0 is a solution of all three
Einstein equations, and, in fact, it does not correspond to a solution of all three Einstein
equations. It is, in fact, the generalization, to the case where u 6= 0, of the line c =
that the trajectories in the (b, c) plane, that corresponded to actual solutions of the
Einstein equations, in the limit u = 0, osculated with, as they switched from the first
to the second branch of the square root, in (194) and (195).
We see, furthermore, that the case α = −3, in (274), satisfies 6c2 − 8 + 4u2 = 0,
to the order given, and thus is the c → 0 limit of this special solution of (270) and
(271), that does not correspond to a solution of all three Einstein equations. We note,
furthermore, that this special solution, of (270) and (271), never rises above the line
, in the (b, c) plane. It in fact approaches this line from below, as b → 0, since
u → 0, as b → 0. Furthermore, when 6c2 − 8 + 4u2 = 0, (270) reduces to du
hence u =
, where, by (272), b2 is the integration constant in (269). Hence
, which does, indeed, solve (271).
Considering, now, the case α = −5
, in (274), we see that 6c2−8+4u2 ≃ 8
c2 near the boundary, hence the square root, R, is nonvanishing, as soon as we move
away from the boundary, so, by (198), this solution will correspond to a solution of
all three Einstein equations. Furthermore, du
starts positive, specifically du
at b = b2, hence u decreases, as b decreases downwards, away from b = b2, hence,
provided du
never becomes negative, and the square root stays real, the square root is
bounded above, by
6c ≃ 2.45c, hence, by (270), we have 3u
1.78u
, and, by (275), we have d
(6c2 − 8 + 4u2) ≤ −
24− 6
6c2 − 8 + 4u2 ≃
−9.30 c
6c2 − 8 + 4u2, hence du
never does become negative, and the square root does
stay real. We can also confirm directly from (271), by considering separately the cases
and c ≤
, that dc
is negative irrespective of the value of u, provided
the square root is real. Furthermore,
≤ u ≤
)1.77
, hence
, hence
d (6c2 − 8 + 4u2)√
6c2 − 8 + 4u2
≤ −9.30c
db ≤ −10.73
db (276)
Hence
6c2 − 8 + 4u2 ≥ 2.68
− arcsin
(277)
With the bound u ≤
)1.77
, this implies that 6c2 − 8 is positive for b
< 0.61,
and is greater than 9.86 for b
= 0.2, by which point u2 < 0.006. Thus this solution
merges into a solution of (204), as b continues to decrease, and for c large compared
, will follow a trajectory of the form (206), in the (b, c) plane, with B
a fixed
number of order 1, that will be the same for all solutions, of this type. Thus we do,
indeed, have a solution of the boundary conditions, such that both a and b are large
compared to κ2/9, at the outer surface of the thick pipe. And moreover, for solutions
of this type, namely with α = −5
in (274), the constant of integration, b2, in (269),
can be identified as b2 = b (y2), the value of b at the outer surface of the thick pipe.
From the behaviour (269), of c near the outer boundary, we see that near the outer
boundary, y2 − y ≃ b2
c, so y tends to a finite value, y2, at the outer
boundary, even though dy
goes to ∞, right at the boundary. y2 will be equal to
a number of order 1, times the value of y at which b̈ vanishes for the interpolating
function (255), on page 109, which by (258), is at y = 0.5220B. Thus the geodesic
distance from the inner surface to the outer surface of the thick pipe is around B.
An alternative method of studying solutions of this type, is to take the ratio of (270)
and (271). Then b cancels out, and we get a single first order differential equation, that
expresses du
, as a function of u and c.
2.5.1 Newton’s constant and the cosmological constant for solutions with
the outer surface in the classical region
We now need to consider whether a solution of this type can fit the observed values
of Newton’s constant, (12), and the cosmological constant, (20). Considering first the
value of Newton’s constant, the current observational limits on extra dimensions in
high energy physics experiments [272], and in measurements of the gravitational force
at short distances [32], imply that the maximum values of y, and b, namely y2, and
b2 = b (y2), are required to be sufficiently small, that a four-dimensional effective field
theory description can be used, for all observations up to the present time. Assuming,
provisionally, that y2 and b2 are, indeed, sufficiently small, the four-dimensional effec-
tive field theory description can be obtained by following the method of Randall and
Sundrum [31].
The first step is to identify the massless gravitational fluctuations about the classical
solution found above. These provide the gravitational fields of the effective theory.
They are the zero-modes of the classical solution, and correspond to replacing the
locally de Sitter metric, gµν , in (94), on page 33, by g̃µν = gµν + hµν , where hµν is a
small perturbation, that, like gµν , depends on position in the four extended dimensions,
but not on y, nor on the coordinates of the compact six-manifold. We note that since
the de Sitter radius of gµν has been set equal to 1, hµν is allowed to very rapidly, with
“Fourier modes” of wavelength down to ∼ 10−29 of the de Sitter radius, corresponding
to the current short distance limit of about a millimetre, on short-distance tests of
Newton’s law, in units of the observed de Sitter radius (22).
The four-dimensional effective theory follows by substituting the zero modes of
the classical solution into the original Hořava-Witten action, (25) plus (28), plus the
analogue of (28) for y2. To determine the value of Newton’s constant, we focus on the
term, in the Einstein action term in (25), that produces the Einstein action, (10), in
four dimensions. The Riemann tensor for the perturbed metric is still given by (96),
on page 34, with gµν replaced by g̃µν , since the derivation of (96) did not make use of
the locally de Sitter property of gµν . I shall denote the metric in eleven dimensions,
with the locally de Sitter metric, gµν , replaced by the perturbed metric, g̃µν , by G̃IJ .
The relevant term, in (25), is then:
G̃µνR τµτν (g̃)
−g̃g̃µνR τµτν (g̃)
dya2b6 (278)
where I replaced 1
, because we are here working in the downstairs picture, and
integrating over only one copy of the bulk, rather than over two copies, one of which is
reflected, as in (25), and I have denoted the coordinates on the compact six-manifold
by zA.
The factor V (M6) =
h is given by (99), on page 36, for a smooth compact
quotient of CH3, and by (100), for a smooth compact quotient of H6. To evaluate the
factor
dya2b6, we note that b is a monotonically increasing function of y, for the
solutions considered in this subsection, so this integral is equal to
a2b6, where
c = db
For the region b1 ≤ b ≤ bq = κ2/9
)0.6551
, we have c ∼ b
, up to a factor of
order 1, and a = A1
, where A1 is related to the integration constant A in the
classical bulk power law (210), on page 90, by (241), on page 104. Thus:
a2b6 ∼
)1.3102τ+3.9306
(τ > −3)
|6+2τ |A
1 (τ < −3)
(279)
up to a factor of order 1. While for the region bq ≤ b ≤ b2, we note that the equations
(270) and (271), on page 113, are invariant under rescaling of b by a constant factor,
so for the general solution, c = f
, where f (1) = 0. Thus the integration constant
B, in the classical bulk power law (206), on page 89, is a fixed number times b2, and
we see from the discussion around (276) and (277) that this fixed number is of order
1. We also note that the classical bulk power law (210), on page 90, for a in terms of
b, will be approximately valid, up to a factor of order 1, right up to the outer surface,
for the solutions considered in this subsection. Thus we find:
a2b6 ∼
)5.4494
)1.3102τ+6.4652
, (280)
up to a factor of order 1, where I used (241).
Comparing (279) and (280), we see that for B ≫ κ2/9, the contribution from the
classical region is large compared to the contribution from the quantum region for all
τ > −4.9345, while for τ ≤ −4.9345, there is no enhancement of the integral for large
We now note that, by definition, the de Sitter radius of the unperturbed metric,
gµν , is equal to 1, so since g̃µν differs from gµν only by a small perturbation, the
use of the metric g̃µν corresponds to measuring distances in units of the de Sitter
radius. We therefore define a rescaled metric ḡµν by ḡµν = (de Sitter radius)
g̃µν ,
which corresponds to measuring distances in ordinary units rather than in units of the
de Sitter radius. Then from (278) and (280), together with (99) or (100), on page
36, and (240), on page 104, we find that for τ > −4.9345, the Einstein action term,
in the four-dimensional effective action, is for the solutions considered in the present
subsection, equal to:
)1.3102τ+6.4652
−ḡḡµνRµτντ (ḡ) (281)
up to a factor of order 1.
Comparing with (10), on page 12, we find that for the solutions considered in the
present subsection, with τ > −4.9345:
)1.3102τ+6.4652
, (282)
up to a factor of order 1. This is the form taken by the ADD mechanism [3, 5], for the
solutions considered in the present subsection. And for τ ≤ −4.9345, we find the same
result, but without the B-dependent factor. Thus for these solutions, there is no ADD
mechanism, unless τ > −4.9345.
Considering, now, the case of TeV-scale gravity, I shall take κ−
9 ≃ 0.2 TeV, so
that κ2/9 ≃ 10−18 metres, as a representative example, which according to Mirabelli,
Perelstein, and Peskin [273] will for six flat extra dimensions be just out of reach at the
Tevatron, but comfortably accessible at the LHC, as I shall review further in subsection
2.6.1, on page 126, and section 5, on page 222. Then from (282) and (12) we find that
)1.3102τ+6.4652
3 ∼ 1032 (283)
for TeV-scale gravity, up to a factor of order 1.
Now as noted above, the bulk power law (210), on page 90, is valid up to a factor
of order 1, for the solutions considered in this subsection, right up to the outer surface,
where b =
2a for these solutions. We also noted that b2 ∼ B, up to a factor of order
1. Thus for these solutions:
)1.7753
, (284)
up to a factor of order 1. Thus from (242), on page 104, we find that for TeV-scale
gravity, the condition for the solutions considered in this subsection to fit the observed
de Sitter radius (22), on page 15, is:
)1.2674−0.6551τ
6 ∼ 1044, (285)
up to a factor of order 1. The solution of (283) and (285) for the minimum value
|χ (M6)| = 1 is
τ = −3.103, B ∼ 1013κ2/9 ∼ 10−5 metres, (286)
and the solution for the maximum value |χ (M6)| ≃ 7× 104 is
τ = −3.023, B ∼ 1013κ2/9 ∼ 10−5 metres. (287)
The value of bq corresponding to (286) and (287) is bq ∼ 108κ2/9 ∼ 10−10 metres, so
the thickness in y of the quantum region is ∼ 20κ2/9 ∼ 10−17 metres.
We note that the value of B in (286) and (287) is about a factor of 10 smaller
than the shortest distance so far studied in precision sub-millimetre tests of Newton’s
law [32]. Nevertheless, we cannot directly conclude that the solutions just obtained
correctly reproduce the d = 4 Newton’s law over any distance even up to the de Sitter
radius ∼ 1026 metres, because we have not fully satisfied the requirement for a valid
reduction to a four-dimensional effective theory, due to the fact that a (y) decreases
from ∼ 1026 metres at the inner surface of the thick pipe, to ∼ 10−5 metres at the outer
surface. Thus any perturbation of gµν , of wavelength less than the de Sitter radius,
will have wavelength less than around B, at the outer surface.
We note that a2 corresponds to the warp factor of the first Randall-Sundrum model
(RS1) [31], and that we live on the “wrong” boundary, from the point of view of the
RS1 model, because the reverse RS1 effect is outweighed by the ADD effect [3, 5],
which is absent from the RS1 model. Arnowitt and Dent [30] have studied Newtonian
forces in the RS1 model, and found that Newton’s law is obtained correctly between
point sources on the RS1 “Planck brane”, which corresponds to the inner surface
of the thick pipe, even though there are problems with Newton’s law between point
sources on the RS1 “TeV brane”. This suggests there is a possibility that Newton’s
law might be obtained correctly for the solutions found in this subsection, even though
the requirement for a valid reduction to a four-dimensional effective theory is not
completely satisfied. However to check this would require repeating the analysis of
Arnowitt and Dent for the solutions found in this subsection, and that will not be done
in this paper.
I shall now consider two alternative ways in which the outer surface of the thick
pipe might be stabilized, consistent with the observed values of Newton’s constant and
the effective d = 4 cosmological constant, for which the value of τ is not fixed uniquely,
and the problem noted above can be avoided. In the first alternative, considered in
the next subsection, a (y) has decreased to around κ2/9 at the outer surface, and there
are Casimir effects near the outer surface. However for τ around the bulk power law
value −0.7753, the main part of the decrease of a (y) takes place in a very narrow part
of the classical region near the outer surface, corresponding to y near 1
= 1.0996B
in the interpolating function (255), on page 109, and in the quantum region near the
outer surface, so that only a fraction ∼ 10−6 of the integral that determines Newton’s
constant comes from values of y for which a (y) is smaller than around 1018 metres.
And in the second alternative, considered in subsection 2.7, on page 137, the outer
surface is stabilized in the classical region by extra fluxes of the three-form gauge field,
whose four-form field strength wraps three-cycles of M6 times the radial dimension
of the thick pipe, and for τ around −0.7753, the value of a (y) at the outer surface is
around 1022 metres.
2.6 Solutions with a as small as κ2/9, at the outer surface of
the thick pipe
I shall now look for solutions of the Einstein equations (162), (163), and (164), on page
81, and the boundary conditions (170) at the outer surface of the thick pipe, such that
the term 4
, in the square root, R, defined in (197), is still extremely small, compared
to the term − 8
, when c = db
is no longer large compared to
, assuming, as in the
preceding two subsections, that the boundary conditions at the inner surface of the
thick pipe have already been solved, such that in the first bulk power law region, we
are on a trajectory (206), (207), and (210), on page 89, with a large value of B
, and
A not yet determined, but such that A
)0.7753
≫ B. We are therefore, in the main
part of the bulk, where both a and b are large compared to κ2/9, and both (y − y1) and
(y2 − y) are large compared to κ2/9, on a solution of the form studied in subsection
2.4.4, on page 107, so that the interpolating function (255), for b as a function of y,
with α given by (257), and the interpolating function (263), for a as a function of b,
are approximately valid, throughout the main part of the bulk.
There is now no possibility of satisfying the boundary conditions at the outer surface
until a has become as small as κ2/9, so that there are Casimir effects on and near the
outer surface. So we are now looking for a solution in which the constant of integration
Ã, in (265), whose value is determined by the boundary conditions at the outer surface,
obtains a very large value in units of κ2/9, by a mechanism analogous to the way in which
the constant of integration B, whose value is determined by the boundary conditions
at the inner surface, can obtain a large value in units of κ
9 , as studied in subsection
2.4.2, on page 93, but with the roles of b and a now reversed.
We therefore now assume that the three observed spatial dimensions, whose curva-
ture has become very large at the outer surface, due to the small size of the warp factor,
a, there, are compact hyperbolic, so as to maximize the available range of dependences
of the Casimir energy densities, at the outer surface, on a. This violates rotational
invariance and Lorentz invariance globally, but not locally. The violation of Lorentz
invariance globally means that the Casimir energy-momentum tensors on and near the
outer surface will not necessarily have the forms (130) and (135), but I shall consider
the case where they do have these forms.
We will find that a large value of Ã
can be obtained self-consistently in the same
way as a large value of B
. By analogy with the region near the inner surface, I shall
first consider the case where t(3) = t(2), as would be appropriate for a dS4 times flat
R7 background. In that case, by analogy with (133), on page 61, the conservation
equation (131), on page 58, reduces to:
dt(2)
t(2) − 4
t(1) = 0 (288)
Hence, in this case, the expansion coefficients C̃(i)n , in the expansions (137), on page
63, are related by:
C̃(1)n = −
(4 + 3n)
C̃(2)n , C̃
n = C̃
n n ≥ 0 (289)
By analogy with (211), on page 94, I shall first consider the case where all the C̃(2)n
are zero, except for a single value of n, and consider the region
∣C̃(2)n
κ2/9 (290)
The Einstein equations (194) and (195), with the lower choice of sign, as appropriate
for this region, and dropping the − 8
and 4
terms in the square root, and the 4
in (195), now become:
= −2c
(n+2)
a8+3n
(291)
(n+2)
a8+3n
2 + n
C̃(2)n
(n+2)
a8+3n
= 0 (292)
The natural independent variable in this region would be a, and we would expect
a trajectory analogous to (214), with ȧ = cda
being given by a power law as a function
of a, with a fixed coefficient, and b being given by a power law as a function of a, with
an undetermined coefficient, analogous to (223). So we try an ansatz:
= −σ̃
, b = B̃
(293)
This implies da
, and c
= − σ̃τ̃
. We then find, from (291), and
(292), that ρ̃ = −6+3n
, which is the same as ρ, at the inner surface of the thick pipe,
and κ
(n+2)
a8+3n
σ̃2τ̃2
. Thus (291) and (292) reduce to
= −2 − 1
σ̃2τ̃ 2
(294)
6 + 3n
− 2− 1
σ̃2τ̃ 2
2 + n
C̃(2)n
σ̃2τ̃ 2
= 0 (295)
which imply 2
C̃(2)n
σ̃2τ̃2
= 10 + 16
, and
(32 + 10n) τ̃ 2 + (32 + 13n) τ̃ + (8 + 4n) = 0 (296)
which has the solutions τ̃ = −1
, and τ̃ = − 4n+8
5n+16
. However, τ̃ = −1
implies, by (294),
that the square root, R, defined in (197), vanishes identically, so we cannot conclude,
from (198), that all three Einstein equations are satisfied, and this also applies to the
solution τ̃ = − 4n+8
5n+16
, when n = 0. Thus I now assume τ̃ = − 4n+8
5n+16
, with n ≥ 1. We
then find:
√− C̃
n (5n+ 16)
6 (15n2 + 96n+ 96)
(297)
We note, from (289), that this requires C̃(1)n to be positive, which corresponds to a
negative contribution to the energy density, T00, which is opposite to the situation at
the inner surface of the thick pipe.
These results can be checked by solving (193) for ḃ
, which gives:
−4 ȧ
κ2t(3)
 (298)
and then using this result, to eliminate ḃ
from the first Einstein equation, (162), which
gives:
κ2t(3) −
5t(1) − 6t(2) − t(3)
= 0 (299)
Comparing with (253), we see that the upper sign in (298) and (299) corresponds to the
lower sign in (194) and (195), and is thus the appropriate sign for the region nearer the
outer surface, in the case under consideration in the present subsection, where a ∼ κ2/9
at the outer surface.
Considering, again, the case where t(3) = t(2), and all the C̃(2)n are zero, except for
a single value of n, and trying for a solution of the form c̃ ≡ ȧ = c̃0 (1 + s), where c̃0 is
the small a power law trajectory found above, and s is a small perturbation, we find,
similarly to the region near the inner surface, that
s = Saη̃, η̃ =
15n2 + 96n+ 96
10n+ 32
(300)
where S is a constant of integration. And since η̃ ≥ 3 for n ≥ 0, trajectories near the
small a power law trajectory tend to converge towards it in the direction of decreasing
a, or in other words, of increasing y, in the sense that s decreases in magnitude in
this direction, so in this sense, the small a power law trajectory is an attractor in the
direction of decreasing a, for n ≥ 0.
Now, by analogy with (218), the small a power law trajectories, found above, can
all be written as:
ȧ2 = −
κ2t(2) (a) −
κ2t(2) (x) (301)
And as near the inner surface of the thick pipe, the integral is convergent, as x → ∞,
because t(2) (x) decreases at least as rapidly as x−8, as x → ∞, although of course
t(2) (x), near the outer surface of the thick pipe, is not the same function as t(2) (x),
near the inner surface of the thick pipe, and (301) does not give an exact solution
of (299), except when t(2) (a) is a pure power of a. However, if t(2) (a) is a linear
combination of two different pure powers, say a−(8+3n) and a−(8+3m), with n and m
large, then the remainder term, in (299), will be ∼ 1√
, while the leading terms will
be ∼ n or m. And for a pure power a−(8+3n), the integral, in (301), is of order 1
compared to the leading term.
Thus, by analogy with the situation near the inner surface of the thick pipe, we
expect that for large −t(2) (a), there will be an attractor trajectory in the (a, ȧ) plane,
such that nearby trajectories approach it, in the direction of decreasing a, or in other
words, in the direction towards the outer surface of the thick pipe, in the sense that
the relative separation of the two trajectories decreases, in the direction of decreasing
a, and this attractor trajectory will approximately be given by:
κ2t(2) (a) (302)
This trajectory will eventually intersect every second bulk power law region bulk power
law trajectory (265), and, as near the inner surface of the thick pipe, we expect each
bulk power law trajectory to curve upwards as it approaches the attractor trajectory,
and then approach the attractor trajectory gradually.
The square roots in (298) and (299) approximately vanish on the approximate small
a attractor trajectory (302), but for n ≥ 1, the small a power law trajectories, which are
approximately described by (302), are already known to be approximate solutions of all
three Einstein equations near the outer surface of the thick pipe, when all the C̃(2)n are
zero, except for a single value of n, and that C̃(2)n is negative, so it seems reasonable to
expect that (302) will also give an approximate solution of all three Einstein equations
in the more general case, when C̃(2)n ≤ 0 for all n ≥ 1.
Now in the same way as in the discussion beginning just before (224), on page 96,
for the region near the inner surface of the thick pipe, we have to consider whether
these solutions can be self-consistent, when we recalculate the expansion coefficients
C̃(i)n in (137), on page 63, for a (y) corresponding to these solutions, in accordance with
the discussion in subsection 2.4.1, on page 90. We first recall that the bulk power law in
the second classical power law region, that corresponds to the bulk power law (206), on
page 89, in the first classical power law region, is (265), on page 110, as we can confirm
from (299) above, with the upper choice of sign. And from the discussion in subsection
2.4.1, a term κ−
9 C̃(i)n
)8+3n
in (137) will be associated with additional terms
9 C̃(i)n,m
)8+3n
c̃m, 1 ≤ m ≤ n, as well as terms with factors of higher derivatives
of a with respect to y, which can, however, be bounded by constant multiples of the
terms without factors of higher derivatives, when the dependence of a on y is by a
power law. The constant of integration Ã, in (265), will have to have a very large
value, in units of κ2/9, in order to fit the observed value (22) of the de Sitter radius, so
the largest additional terms will be those with m = n.
Thus the Casimir terms in (298) and (299), with the upper choice of sign, will
be significant near the outer surface for κ
2/9c̃
≥ 1, which from (265) corresponds to
)0.5326
. Defining ãq to be the value of a where this is an equality, we then
find, from (266), that:
)0.5326
= 0.03757
)1.2427
(303)
And defining b̃q to be the corresponding value of b, we find from (254), on page 109,
and (264), on page 110, that
= 1.6884
)1.1450
(304)
Then in the same way as in subsection 2.4.2, on page 93, for the region near the
inner surface, we find that the only self-consistent way to obtain a large value of the
integration constant Ã, is for c̃ to depend linearly on a in the quantum region a ≤ ãq,
which results in (291) and (292) for n = −8
, and an effective coefficient C̃
, for
ãq ≥ a≫ κ2/9. This results self-consistently in (293), with ρ̃ = τ̃ = 1, and σ̃ = 13
so that the effective coefficient C̃
has to be > 0 in order to obtain the linear relation.
We note that η̃, in (300), takes the value −10 when n = −8
, so the linear trajectory
is a very strong attractor in the direction of increasing a. However, in the same way
as for the corresponding result for the region near the inner surface, this has not taken
account of the fact that in the presence of deviations from the linear trajectory, the
equations to be solved will no longer be precisely (291) and (292), with n = −8
Now a continues to decrease with increasing y in the quantum region near the outer
surface, since c̃ = da
is still negative in the quantum region. Hence since b
has the
fixed value B̃
in the quantum region, b stops increasing with increasing y at the
upper limit b̃q of the classical region, and decreases with increasing y in the quantum
region. Thus a necessary condition for the existence of a solution to the boundary
conditions at the outer surface is that b must be comparable to or larger than a at
the start of the quantum region, or in other words, ãq ≫ b̃q must not hold, for if ãq
was ≫ b̃q, the boundary conditions at the outer surface would not depend significantly
on the integration constant Ã, so that Ã would be undetermined, and B would be
overdetermined.
In the next subsection, I shall determine the values of B and Ã required to fit the
observed values of Newton’s constant and the cosmological constant, for given values
of τ and κ−
9 , assuming that this consistency condition is satisfied. We will then find
that for τ = 1, which follows from assuming that t(3) = t(1) in the quantum region
near the inner surface of the thick pipe, the consistency condition cannot be satisfied
unless κ−
9 is much smaller than the minimum value ∼ 0.1 TeV allowed by current
observations. However the linear relation between b and a in the quantum region near
the outer surface, which follows from (293) and (296) for n = −8
, on rejecting the
solution τ̃ = −1
, is a consequence of the assumption that t(3) = t(2) near the outer
surface, and there is no reason to expect this relation to be valid when a and b depend
exponentially on y.
Thus in a similar way to the discussion following (231), on page 100, we should
discard the assumption that t(3) = t(2) near the outer surface, and assume that the
t(i), in (130), on page 58, are constrained only by the conservation equation (131). We
would then expect, by analogy with the situation near the inner surface, that almost
any value of τ̃ could be obtained, provided there exists a suitable smooth compact
quotient M3 of H3, and a choice of spin structure on M3, that results self-consistently
in the appropriate values of the independent coefficients C̃
and C̃
2.6.1 Newton’s constant and the cosmological constant
I shall now determine the values of the integration constants B, in (255), on page 109,
and A, in (263), or equivalently, Ã, in (265), and the constants τ , in (233), on page 101,
and κ2/9, for which the solutions found above can fit the observed values of Newton’s
constant, (12), and the cosmological constant, (20), and check that this type of solution
is consistent with observational limits on the existence of large extra dimensions, and
can avoid the possible problem noted in the discussion following (287), on page 118,
for the solutions studied in subsection 2.5, on page 111. I shall then check that this
type of solution is consistent with experimental limits on deviations from Newton’s
law at sub-millimetre distances, in subsection 2.6.2, on page 132, and with precision
solar system tests of General Relativity, in subsection 2.6.3, on page 135. Some further
consequences of the warp factor decreasing to a small value, at the outer surface of the
thick pipe, in this type of solution, are considered briefly in subsection 2.6.4, on page
We can follow the same method as used in subsection 2.5.1, on page 115. The term,
in the Einstein action term in (25), that produces the Einstein action, (10), in four
dimensions, is again given by (278), where b (y) is now given by (255), and a, as a
function of b, is given by (263). Thus we now have:
a2b6 ≃ A2B6
)1.5506
(305)
where f (Y ) is defined by:
f (Y ) ≡ 5.1220 (1− 0.9094Y )
0.0651
Y 1.5346
(1− 0.9094Y )0.3549 + 12.0816Y 0.8448
(306)
The function f (Y ) is illustrated in Figure 2. The peak is at Y = 0.5777, at
which point the value of the function is 0.03002. The function is 0 at Y = 0, and at
Y = 1.0996, and by use of PARI/GP [274], we find:
∫ 1.0996
dY f (Y ) = 0.02967 (307)
0.0 0.5 1.0
Figure 2: The function f (Y ) defined in (306)
The contribution to this integral, from the regions 0 ≤ Y ≤ y1
, and y2
≤ Y ≤ 1.0996,
will be negligible, to the accuracy to which we are working, so we now find:
dya2b6 ≃ 0.02967κ
)5.4494
= 0.02967κ
9 A21
)1.3102τ+6.4652
(308)
instead of (280). The numerical coefficient in (308) should now be approximately
correct, for the solutions found in subsection 2.6, on page 120, to the extent that
the interpolating functions (255), and (263), are approximately valid, whereas the
numerical coefficient, in (280), was only valid up to a factor of order 1.
The result (308) is for a smooth compact quotient of CH3. To obtain the cor-
responding result for a smooth compact quotient of H6, we note that throughout
the range where they give significant contributions to the integral, a and b are solu-
tions of the vacuum Einstein equations, and a is so large that the curvature of the
four-dimensional de Sitter space can be neglected. We recall that we have chosen
the metric hAB for H
6 to have radius of curvature equal to 1, so that RABCD (h) =
hAChBD−hADhBC , and RAB (h) = 5hAB, as stated after (97), on page 34. Then looking
at the Ricci tensor components (97), and noting that RAB (h) = 4hAB for the stan-
dard metric on CH3 introduced in subsection 2.2, on page 25, we see that the vacuum
Einstein equations for CH3, when a is so large that the curvature of the dS4 can be
neglected, can be transformed into the corresponding equations for H6, by rescaling y
by a factor
. Furthermore, derivatives with respect to y are larger for H6 than for
CH3 by a factor
, so the range of y is smaller for H6 than for CH3, by a factor
. Thus the integral
dya2b6 for H6 is obtained from the corresponding integral
for CH3 by multiplying by a factor
, or in other words, replacing the coefficient
0.02967, in (308), by 0.02654.
The integral over the compact six-manifold, in terms of the Euler number of the
compact six-manifold, will be the same as before, so we find that when the compact
six-manifold is a smooth compact quotient of CH3, the Einstein action term, in the
four-dimensional effective action, for the solutions considered in subsection 2.6, will be
equal to:
0.3067
)5.4494
−g̃g̃µνRµτντ (g̃) (309)
And when the compact six-manifold is a smooth compact quotient of H6, we get the
same result as in (309), but with the numerical coefficient replaced by
×0.3067 ≃
0.4389.
Thus from the relation (242), on page 104, between A, and the observed de Sitter
radius (22), and the discussion following (281), on page 117, we see that when we define
the rescaled metric ḡµν by ḡµν = (de Sitter radius)
g̃µν as before, so as to measure
distances in ordinary units, rather than in units of the de Sitter radius, the Einstein
action term, in the four-dimensional effective action, for the solutions considered in
subsection 2.6, will for smooth compact quotients of CH3 be equal to:
− 0.3067
1.27722τ
)1.3102τ+6.4652
−ḡḡµνRµτντ (ḡ) (310)
And for smooth compact quotients of H6, we get the same result as in (310), but with
the numerical coefficient replaced by 0.4389
1.18092τ
Thus, comparing with (10), we find that for smooth compact quotients of CH3:
≃ 15.416
1.27722τ
)1.3102τ+6.4652
(311)
And for smooth compact quotients of H6, the numerical coefficient is replaced by
22.062
1.18092τ
. This is the form taken by the ADD mechanism [3, 5], for the solutions consid-
ered in subsection 2.6, on page 120. We see that in the same way as for the solutions
considered in subsection 2.5, on page 111, there is no ADD effect unless τ > −4.9345.
This is due to the fact that for the classical region in the bulk, and for τ < 0, also
for the quantum region near the inner surface of the thick pipe, we live on the wrong
boundary, from the point of view of the first Randall-Sundrum model [31], and for
τ < −4.9345, the reverse RS1 effect outweighs the ADD effect.
For τ = 1, we find from (311) that for smooth compact quotients of CH3:
≃ 0.7491
|χ (M6)|0.1715
0.1286
(312)
Considering, now, the case of TeV-scale gravity, we will find in section 5, on
page 222, that κ is related to the gravitational masses M , Mp, and MD, with D = 11,
defined respectively by Mirabelli, Perelstein, and Peskin [273], Giddings and Thomas
[275], and Giudice, Rattazzi, and Wells [11], by M =Mp = 2
9MD = 2π
. I shall
use the results of Mirabelli, Perelstein, and Peskin, for six flat extra dimensions, as an
indication of the current experimental limits on κ−
9 . Thus from their Table 1, we see
that in 1998, the LEP 2 lower bound on κ−
9 was around 107 GeV, and the Tevatron
lower bound was around 125 GeV. And the final lower bound on κ−
9 attainable at
the Tevatron is expected to be around 166 GeV, and the final lower bound on κ−
attainable at the LHC is expected to be around 677 GeV.
As a representative example of TeV-scale gravity, I shall consider the case where
the Giudice, Rattazzi, and Wells gravitational massMD, for D = 11, is equal to 1 TeV,
which corresponds to κ−
9 = 0.2217 TeV, so that κ2/9 = 8.899×10−19 metres. We then
find from (12), on page 13, that for τ = 1, and smooth compact quotients of CH3:
B ≃ 1.515× 10
|χ (M6)|0.1715
κ2/9 ≃ 1.348× 10
−14 metres
|χ (M6)|0.1715
(313)
Thus from (242), on page 104, and (22), on page 15, we have for τ = 1, and smooth
compact quotients of CH3:
A ≃ 8.6× 10
30 metres
|χ (M6)|0.0328
≃ 5.4× 10
|χ (M6)|0.0328
(314)
Thus from (266), on page 110, the integration constant Ã, in (265), is given for τ = 1,
and smooth compact quotients of CH3, by:
Ã ≃ 3.15× 1061
0.3386
metres ≃ 3.54× 1079
0.3386
κ2/9 (315)
Thus from (303) and (304), on page 124, we find that for τ = 1, and smooth compact
quotients of CH3:
≃ 2.33× 1042
0.1803
1.033× 105
|χ (M6)|0.1964
(316)
Thus the consistency requirement that when the exponent τ̃ in (293), on page 121,
is equal to 1, ãq must not be large compared to b̃q, is violated for τ = 1. Thus the
relation t(3) = t(1) near the inner surface of the thick pipe and the relation t(3) = t(2)
near the outer surface cannot both be satisfied, but as noted in subsections 2.4.2, on
page 93, and 2.6, on page 120, there is no reason for either of these relations to be
satisfied, since a and b depend exponentially on y in the quantum regions. If τ̃ is < 0,
there is no consistency condition on ãq and b̃q, since b (y) continues to increase with
increasing y in the quantum region near the outer surface.
When τ = −0.7753, as for the classical power law (210), on page 90, in the first
classical region, we find from (311) that for smooth compact quotients of CH3:
0.5646
|χ (M6)|0.1361
0.1835
(317)
And for smooth compact quotients of H6, the coefficient 0.5646 is replaced by 0.5406.
Considering, again, the case of TeV-scale gravity, with κ−
9 = 0.2217 TeV, we find
that for τ = −0.7753, and smooth compact quotients of CH3:
B ≃ 7.864× 10
|χ (M6)|0.1361
κ2/9 ≃ 6.998× 10
−13 metres
|χ (M6)|0.1361
, (318)
1.83× 1026 metres
|χ (M6)|0.1292
1.14× 1061
|χ (M6)|0.1292
, (319)
Ã ≃ 5.29× 1048
0.0750
metres = 5.95× 1066
0.0750
κ2/9. (320)
Comparing (315) and (320), we see that the cost of decreasing B
by a factor of
around 50, by increasing τ from −0.7753 to 1, is to increase Ã
by a factor of around
6 × 1012. Thus it does not seem likely that B will be much smaller than the value
(318) corresponding to τ = −0.7753. Thus from the upper bound of around 7 × 104
on |χ (M6)| found in subsection 2.3.6, on page 66, it does not seem likely that B
be much smaller than 105, for TeV-scale gravity.
By decreasing τ below −0.7753, it will be possible to decrease Ã
at a cost of
increasing B
, until as τ approaches the values near −3 in (286) and (287), on page
118, the assumption made in subsection 2.6, on page 120, that the term 4
in the
square root R defined in (197), on page 87, is still extremely small compared to the
term − 8
, when c = db
is no longer large compared to
, will no longer be valid, and
the type of solution considered in subsection 2.6 will resemble the solutions studied in
subsection 2.5, on page 111, except in the region close to the outer surface. However
a will still decrease to around κ2/9 at the outer surface for the solutions considered in
subsection 2.6, because there are no solutions of the Einstein equations where c goes
to zero at a finite value of b on the second branch of the square root. For if such a
solution existed, then expanding u = b
near the boundary as in subsection 2.5, we
would find an equation that is obtained from (273), on page 113, by reversing the sign
of the square root. This leads to the same quadratic equation as before, with the same
solutions, (274), as before. But the solution α = −3 corresponds to the osculating line,
which is not a solution of all three Einstein equations, and the solution α = −5
longer solves the original equation.
The large values of B
and Ã
are the large numbers built into the structure of
the universe, that make the universe into the stiff, strong structure that we observe.
We note that due to the unique properties of smooth compact quotients of H3, it might
be easier to obtain large values of Ã
than of B
. The three-volume V (M3) of a
compact hyperbolic three-manifold M3 is a topological invariant when the Ricci scalar
has a fixed value, which is usually chosen to be 6, corresponding to sectional curvature
equal to −1. And uniquely to three dimensions, for any given three-volume V1, there is
a finite, larger three-volume V2, such that there are an infinite number of topologically
distinct compact hyperbolic three-manifolds M3 with Ricci scalar equal to 6, such
that V1 ≤ V (M3) ≤ V2. The existence of this property follows from a construction
of Thurston [276], and its uniqueness to three dimensions follows from a theorem of
Wang [277], as I shall briefly discuss in section 3, on page 160.
There is no observational upper limit to the topological invariant V (M3). For
approximately homogeneous M3 the Casimir terms in the energy-momentum tensor
near the outer surface of the thick pipe may tend to become independent of the topology
of M3 for large V (M3), but all but a finite number of the M3 with volumes in a finite
range V1 to V2 produced by the Thurston construction are significantly inhomogeneous.
The inhomogeneity takes the form of a finite number of finite length “spikes” with
smooth rounded ends, that approximate the infinite length “cusps” of the finite volume
non-compact quotients of H3 to which the smooth compact quotients of H3 produced
by the Thurston construction are related. The value of Ã
depends only on the
average over M3 of the functions t(i), and it would seem reasonable to expect that for
the majority of the smooth compact M3 produced by the Thurston construction, these
averages will continue to depend on the topology of M3 for arbitrarily large V (M3),
and perhaps might tend to populate some ranges of values densely.
Comparing (318) with (320), we see that when τ has the value −0.7753, correspond-
ing to the classical power law (210) in the first classical region, the value of B
required
for TeV-scale gravity is relatively small in comparison to the very large value required
for Ã
. Moreover, from (242), (22), (317), and (266), we find that for τ = −0.7753 and
smooth compact quotients of CH3, with a general value of κ2/9:
≃ 3.53× 10112
)1.3670
0.0750
(321)
Thus for τ = −0.7753 the required value of Ã
is minimized by choosing κ2/9 as large as
possible, which means TeV-scale gravity, provided this is consistent with the precision
tests of Newton’s law down to sub-millimetre distances [32], which I will check in the
next subsection.
2.6.2 Comparison with sub-millimetre tests of Newton’s law
We now need to check that TeV-scale gravity, in the type of model considered here, is
consistent with the precision tests of Newton’s law, down to sub-millimetre distances.
The shortest distance over which Newton’s law has been tested precisely is currently
about 0.2 millimetres, so to be sure of the validity of the four dimensional effective
action description, we require that for all y that make a significant contribution to the
integral (308),
a(y1)
times 0.2 millimetres is large compared to both y and b (y). From
subsection 2.5.1, on page 115, we know that if τ > −4.9345, so that there is an ADD
effect, then the dominant contribution to the integral on the first branch of the square
root comes from the classical region.
I shall consider the case where τ = −0.7753, as in the classical power law (210), on
page 90, in the first part of the classical region, and τ̃ = −0.3101, corresponding to the
classical power law (254), on page 109, in the second part of the classical region. Then
the interpolating function (263), on page 110, will be approximately valid throughout
the whole range from the inner surface to the outer surface of the thick pipe. I shall
make the approximation of treating the interpolating function (255), on page 109, as
if it was also valid throughout the whole range from the inner surface to the outer
surface.
The condition to be sure of the validity of the four dimensional effective action
description will be strictest as y approaches the outer surface of the thick pipe, at
y ≃ 1.0996B, since a (y) decreases monotonically with increasing y, and b (y) increases
monotonically with increasing y. Moreover, we see, from (255), that b (y) is comparable
to y in the mid-region of the thick pipe, but becomes large compared to y, as either
boundary of the thick pipe is approached. Thus it is sufficient to check the requirement
for b (y), in the region where b is approaching the outer surface of the thick pipe. We
then have, from (255), (263), and (242), that:
b ≃ 1.5123B
1.0996B − y
)0.1449
(322)
a ≃ 0.2459
|χ (M6)|0.1292
)2.4495
)3.2247
× de Sitter radius (323)
Thus, by (323), the requirement is that for all y that make a significant contribution
to the integral (308):
0.2459
|χ (M6)|0.1292
)2.4495
× 0.2 millimetre
)4.2247
(324)
And for TeV-scale gravity, this becomes, by (313):
1.89× 103
0.1124
)4.2247
(325)
And by (322), this becomes:
1.0996−
7.72× 10−5
|χ (M6)|0.1836
(326)
Now |χ (M6)| ≥ 1, hence (326) will be satisfied, provided:
1.0996−
≫ 7.72× 10−5 (327)
Now the contribution to the integral (307), from the region where Y is within 7.72×10−5
of the upper limit, is 1.35 × 10−6, which is a fraction 4.54 × 10−5 of the full integral
(307). Thus for tests of Newton’s law at distances around 0.2 millimetres, we anticipate
deviations from Newton’s law, in the shape of a small change in the effective value of
Newton’s constant, at the level of about 50 parts in a million, or 5× 10−3 percent.
To compare this result with the measurements of Hoyle et al [32], we note that
one of the ways they expressed their results, was by giving 95% confidence level limits
on the magnitude of the parameter α, as a function of λ, in a modified Newtonian
potential of the form:
V (r) = −Gm1m2
1 + αe−r/λ
(328)
The 95% confidence level limits on |α|, as a function of λ, are given in their Table
XIII, from which we see that for λ = 0.10 millimetre, |α| ≤ 1.8 × 101. For λ = 0.25
millimetre, |α| ≤ 4.3×10−1. And for λ in the range 1.00 millimetres to 10.0 millimetres,
the upper bound on |α| is around 10−2.
The form of equation (328) is such that for r large compared to λ, the correction
to Newton’s law is negligible, but for r comparable with λ, or smaller than λ, there
is effectively a modification of Newton’s constant, by a factor ∼ (1 + α). Thus for
the form of TeV-scale gravity considered in the present paper, the expected deviations
from Newton’s law, at distances around a millimetre, are around 5×10−3 times smaller
than the current best experimental limits of Hoyle et al.
It is interesting to note that the upper bound, (326), on y, for the four dimensional
reduction to be valid, for submillimetre tests of Newton’s law, from the inner surface
of the thick pipe, up to y, corresponds, by (313), to:
1.0996B − y ≫
5.40× 10−17 metres
|χ (M6)|0.3197
(329)
or in other words, since κ2/9 = 8.899× 10−19 metres, for TeV-scale gravity, to:
1.0996B − y ≫ 60.7
|χ (M6)|0.3197
κ2/9 (330)
On the other hand, from (303), on page 124, the value of
that corresponds to
the value (320) of Ã is 3.67× 1035 |χ (M6)|0.0399, so the thickness in y of the quantum
region near the outer surface is around κ2/9 ln
≃ 81.9κ2/9. Thus if the precision
of the submillimetre tests of Newton’s law could be increased by another three decimal
places, they would be probing the quantum region near the outer surface of the thick
pipe, for the solutions considered in subsection 2.6, when τ = −0.7753.
From (322) and (318), the value of b, at the value of y where the inequalities in
(329) and (330) become equality, is:
b ≃ 5.96
0.0266
B ≃ 4.17× 10
−12 metres
|χ (M6)|0.1095
≃ 4.69× 10
|χ (M6)|0.1095
κ2/9 (331)
And from (323) and (22), the corresponding value of a is:
a ≃ 2.09× 10
|χ (M6)|0.1095
×de Sitter radius = 3.16× 10
18 metres
|χ (M6)|0.1095
3.55× 1036
|χ (M6)|0.1095
κ2/9 (332)
2.6.3 Comparison with precision solar system tests of General Relativity
There are also very precise tests of General Relativity, via lunar laser ranging mea-
surements of the lunar orbit, using reflectors left on the surface of the moon by Apollo
astronauts, and by unmanned Soviet lunar missions [278, 279]. In particular, a test of
the equivalence principle, obtained from a fit of lunar laser ranging data, gives a value
for the difference in the ratio of gravitational mass to inertial mass, MG/MI , between
the Earth (e) and the Moon (m). The value quoted in [278], which has been corrected
for solar radiation pressure, is:
= (−1.0± 1.4)× 10−13 (333)
To check the consistency with this measurement, of the models studied here, we need
to decide, if this measurement is interpreted as giving a bound on the variation of
Newton’s constant with distance, what the shortest relevant distance is. The lunar
orbit is determined by the gravitational interaction between the Moon and the Earth,
while both move in the gravitational field of the Sun (s).
From equation (2) of [278], the effective acceleration of the Moon with respect to
the Earth, ~a = ~am − ~ae, for the three-body Earth-Moon-Sun system, is:
~a = −GN
−GNMs
+GNMs
(334)
The last two terms in (334) represent the solar effect on the motion of the Moon with
respect to the Earth. A violation of the equivalence principle would produce a lunar
orbit perturbation proportional to the difference in the two MG/MI ratios.
From the form of equation (334), it appears that a small percentage difference inGN ,
between the first term, and the last two terms, corresponding to a small percentage
difference in GN , for the Earth-Moon distance, and the Earth-Sun distance, might
result in an orbital perturbation different in form, but of the same order of magnitude,
as the perturbation resulting from a similar percentage difference in the two MG/MI
ratios. Thus I shall provisionally interpret the measurement (333), as also giving an
order of magnitude bound on the percentage difference of Newton’s constant for the
Earth-Moon distance, and for the Earth-Sun distance. Thus we have to repeat the
calculation performed above, for tests of Newton’s law over distances of around 0.2
millimetres, for distances around the Earth-Moon distance, which is around 4 × 108
metres. Instead of (327), we now find that a sufficient condition on y, for the four-
dimensional reduction to be valid from the inner surface of the thick pipe up to y, is
that:
1.0996−
≫ 6.20× 10−25 (335)
It follows immediately from the flat topped shape of the function f (Y ), Figure 2,
together with the fact that the peak of the function is outside the range excluded by
(335), that the contribution to the integral (307), from the region excluded by (335),
is not more than a fraction ∼ 10−23 of the value of the integral, if ≫ is interpreted
as meaning larger by a factor of at least 10. Thus for the form of TeV-scale gravity
considered in the present paper, the fractional difference of GN for the Earth-Moon
distance, from GN for the Earth-Sun distance, will not be more than around 10
at the most, which is smaller than the bound given by (333), interpreted as discussed
above, by a factor of around 10−10.
Thus, notwithstanding the remarkable precision of the lunar laser ranging measure-
ments, the submillimetre tests of Newton’s law are currently closer to testing the form
of TeV-scale gravity considered in subsection 2.6, with τ = −0.7753.
2.6.4 Further consequences of the warp factor decreasing to a small value,
at the outer surface of the thick pipe
The fact that the warp factor, a2 (y), decreases to a small value as y approaches y2, in
the solutions considered in subsection 2.6, implies that there are short spacelike paths
through the bulk between points that are separated by large distances in the observed
universe. However, for the solutions considered in the present paper, it is not possible,
even in principle, to send signals through the bulk to distant parts of the observed
universe, at what would appear to be superluminal speeds, from the point of view of
observers on the inner surface of the thick pipe, because the time dimension scales with
exactly the same scale factor, a (y), as the three observed spatial dimensions.
It would be interesting to find out whether or not this conclusion could be modified
in cosmological-type solutions, which would require the analysis of some coupled partial
differential equations, with the time, and y, as independent variables. In particular,
it would be interesting to find out whether or not an effect of this type could provide
an alternative to inflation, for solving the horizon problem of the early universe [78].
Alternative solutions to the horizon problem, of this type, have been discussed in
[280, 281, 282, 283, 284, 285, 286, 287, 288]. It would also be interesting to find out
whether or not an effect of this type would be consistent with the type of causality
constraints recently discussed by Arkani-Hamed et al [289]. However these questions
will not be addressed in the present paper.
2.7 Stiffening by fluxes wrapping three-cycles of the compact
six-manifold times the radial dimension
The occurrence of non-vanishing fluxes of form fields in the de Sitter backgrounds for
type IIB superstrings constructed by Kachru, Kallosh, Linde, and Trivedi [290] suggests
that it might also be interesting to consider solutions with extra fluxes of the four-form
field strength of the three-form gauge field in the present context, so I shall now consider
the possible effects of fluxes wrapping three cycles of the compact six-manifold, times
the Hořava-Witten one-cycle along the eleventh dimension, in the upstairs picture. I
shall assume, to start with, that there will not be enough non-vanishing components
of the three-form gauge field, for the non-linear term in the classical field equation for
the three-form gauge field to be non-vanishing, so that we can treat the classical field
equation for the three-form gauge field as a linear equation, and add solutions. We
then seek a classical solution, such that only the components GABCy are non-zero, and
GABCy (z, y), where z denotes the coordinates on the compact six-manifold, has the
factorized form
GABCy (z, y) = GABC (z) f (y) (336)
Now the Bianchi identity reads:
∂IGJKLM + ∂JGKLMI + ∂KGLMIJ + ∂LGMIJK + ∂MGIJKL = 0 (337)
With the ansatz (336), one component of this reads:
(∂AGBCD (z)− ∂BGCDA (z) + ∂CGDAB (z)− ∂DGABC (z)) f (y) = 0 (338)
which, since f (y) 6= 0 by assumption, is the Bianchi identity for the three-form factor
GABC (z).
Now when the gravitino field vanishes, the classical field equation for the three-form
gauge field CIJK , from the action (25), is:
−GGIMGJNGKOGLPGMNOP
−GGJKLI4...I7I8...I11GI4...I7GI8...I11 = 0
(339)
where the metric in eleven dimensions is denoted GIJ , as in (94), so that G
I1I2...I11 de-
notes the tensor 1√−Gǫ
I1I2...I11. Let us now assume that GIJKL is zero, if any component
is along the four observed dimensions. Then there are at most seven possible values
for each index, such that GIJKL is non-zero, so the term in (339) bilinear in GIJKL
vanishes, and the field equation reduces to:
−GGIMGJNGKOGLPGMNOP
= 0 (340)
Now, bearing in mind the metric ansatz (94), one set of components of this equation,
for the factorized ansatz (336), reads:
hhADhBEhCFGDEF (z)
a (y)
f (y) = 0 (341)
Now, since
−g, a (y), and f (y) are assumed to be non-vanishing, this equation,
together with (338), implies that GABC (z) is a Hodge - de Rham harmonic three-form
on the compact six-manifold. So by standard Hodge - de Rham theory, there are B3
linearly independent solutions GABC (z) of (338) and (341), where B3 is the third Betti
number of the compact six-manifold. I shall now assume that GABC (z) is a Hodge -
de Rham harmonic three-form on the compact six-manifold.
The remaining set of components of (340), that are not satisfied identically for the
factorized ansatz (336), are:
a (y)
f (y)
hhADhBEhCFGDEF (z) = 0 (342)
Thus f (y) is equal to a fixed number, times a (y)
, so, absorbing the fixed number
into GABC (z), we find that:
GABCy (z, y) = GABC (z) a (y)
(343)
where GABC (z) is a Hodge - de Rham harmonic three-form on the compact six-
manifold. We note that (343) applies for all y, in the upstairs picture, since under reflec-
tion in the orbifold hyperplane at y = y1, we have GUVWy (x, 2y1 − y) = GUVWy (x, y),
and also a (2y1 − y) = a (y), where x here denotes the coordinates on M10.
Now, following an argument given by Witten, in section 2 of [151], we consider a
four-cycle X in M10, on the y1+ side of the orbifold hyperplane at y = y1, and apply
the relation (44), with the substitution (47). If the Pontryagin number of X is zero,
then the RR term in (47) will not contribute to the integral of the right-hand side of
(47) over X , so we find that
G|y=y1+ =
dxUdxV dxWdxX GUVWX |y=y1+ (344)
is equal to 1
X trF
(1) ∧ F (1), which Witten indicates is a four-dimensional charac-
teristic class of the E8 bundle at y1, and is equal to an integer. However, (344) is a
topological invariant for smoothly varying G, and thus has the same value no matter
what value of y it is evaluated at, and, indeed, has the same value for any four-cycle
in M11 = M10 × S1/Z2 that is topologically equivalent to X .
Following Witten’s argument, if we now consider Hořava-Witten theory with a
large value of (y2 − y1), specifically, much larger than the diameter of X , and the
integral (344) at some value of y a long distance away from both y1 and y2, then it
would seem unlikely that the value of the integral would depend on whether or not
there exist orbifold hyperplanes a very large distance away, at y1 and y2. Thus we
expect that (344) should be equal to an integer for an arbitrary four-cycle X with zero
Pontryagin number, for smoothly varying G, in supergravity in eleven dimensions.
In other words, (344) gives a form of Dirac quantization condition on the integral
of the Cremmer-Julia-Scherk four-form field strength G, over a four-cycle with zero
Pontryagin number. Witten gives further arguments supporting this interpretation,
and also, a generalization of the quantization condition, to four-cycles with non-zero
Pontryagin number.
Witten’s arguments do not cover the case of a four-cycle, in the upstairs formulation
of Hořava-Witten theory, that has the form of a three-cycle in M10, times a one-cycle
that wraps the S1 in the y direction. However, since the Hořava-Witten boundary
conditions, at the orbifold fixed-point hyperplanes, imply that GUVWy is continuous
across the orbifold fixed-point hyperplanes, and such a four-cycle automatically has
zero Pontryagin number, I shall assume that (344) also has an integer value, for such
a four-cycle, and that this applies, in particular, for the factorized ansatz (336). Thus,
from (343), we find that, for any three-cycle, Z, of the compact six-manifold:
dzAdzBdzCGABC (z)
dya (y)
(345)
must be equal to an integer.
We now have to calculate the modified value of the contribution (158), on page 79,
of the three-form gauge field to the energy-momentum tensor, (14), in the presence of
fluxes wrapping three-cycles of the compact six-manifold, with the ansatz (336). Since
the three-form field configurations considered in subsection 2.3.8 are only significant,
in the energy-momentum tensor, near the inner surface of the thick pipe, while, from
(343), the three-form field configurations considered in the present subsection are sup-
pressed by the very small factor a (y)
, near the inner surface of the thick pipe, I
shall provisionally assume that cross terms in the energy-momentum tensor, between
the three-form field configurations considered in subsection 2.3.8, and those considered
in the present subsection, can be neglected, for compactifications on smooth compact
quotients of CH3, while for compactifications on smooth compact quotients of H6, the
three-form field configurations of the type considered in subsection 2.3.8 are absent,
since Witten’s topological constraint is satisfied with zero G, as noted at the end of
subsection 2.3.7.
From the metric ansatz (94), we find:
GKNGLOGMPGAKLMGBNOP =
hCEhDFGACD (z)GBEF (z) (346)
I shall now assume, as in the study of the Casimir contributions to the energy-
momentum tensor in subsection 2.3.4, that the Einstein equations are expanded in
harmonics on the compact six-manifold, following the procedure of Lukas, Ovrut, and
Waldram [67], and I shall consider the Einstein equations in the approximation of drop-
ping all but the lowest harmonic. I shall also assume that GABC (z), which is a sum of
constant multiples of B3 linearly independent Hodge - de Rham harmonic three-forms,
where B3 is the third Betti number of the compact six-manifold, has been chosen such
hhCEhDFGACD (z)GBEF (z) =
CFhDGhEHGCDE (z)GFGH (z) (347)
hhABh
CFhDGhEHGCDE (z)GFGH (z) = G
hhAB (348)
for a suitable real constant G > 0. These conditions (347) and (348) constitute at
most 20 + 20 linearly independent constraints on the B3 independent coefficients in
GABC (z), and thus can presumably always be satisfied, for sufficiently large B3, unless
this somehow conflicts with the requirement that (345) be an integer for all three-cycles
Z, which I shall assume does not occur. Then in the approximation of dropping all
but the lowest harmonic, (346) becomes:
GKNGLOGMPGAKLMGBNOP =
2b6a8
2 (349)
Similarly, we find:
GKNGLOGMPGyKLMGyNOP =
hADhBEhCFGABC (z)GDEF (z) (350)
And in the approximation of dropping all but the lowest harmonic, this becomes:
GKNGLOGMPGyKLMGyNOP =
G̃2 (351)
where the real constant G̃ > 0 is defined by
hhADhBEhCFGABC (z)GDEF (z) = G̃
h (352)
We also find:
GQRGKNGLOGMPGQKLMGRNOP =
hADhBEhCFGABC (z)GDEF (z) (353)
Thus, in the approximation of dropping all but the leading harmonic, we have:
QRGKNGLOGMPGQKLMGRNOP =
G2GAB (354)
QRGKNGLOGMPGQKLMGRNOP =
G̃2Gµν (355)
QRGKNGLOGMPGQKLMGRNOP =
G̃2 (356)
Hence, from (158), we find the following additional contributions to the energy-
momentum tensor of the three-form gauge field, to be added to (159), in the approx-
imation of neglecting cross terms between the three-form gauge field configurations
considered in subsection 2.3.8, and those considered in the present subsection:
T (3f)µν = −
12κ2b6a8
Gµν , T
AB = 0, T
12κ2b6a8
(357)
These components satisfy the conservation equation (131), with t(1), t(2), and t(3),
defined by (130), now interpreted as unrestricted functions of y.
2.7.1 The region near the outer surface
Now comparing with (130) on page 58, and with (194) and (195), on page 87, with
the upper choice of sign for the square root, we see that the new term in the square
root, resulting from (357), has the correct sign, namely the same sign as the 4
term,
to make possible a solution of the boundary conditions, at the outer surface of the
thick pipe, with both a and b large compared to κ2/9 at the outer surface, by the same
mechanism as in subsection 2.5, on page 111. Furthermore, from (210), on page 90,
we see that 1
behaves as b0.2024 in the first bulk type power law region, governed by
(205), on page 89, and (209), on page 90. Thus the correction terms from (357) do
indeed grow in importance with increasing b, or equivalently, with increasing y, and
thus the most important of the correction terms, which is the one in the square root,
has the correct qualitative behaviour, for a suitable value of G̃, to make possible a
solution of the boundary conditions at the outer surface in the classical region, along
the lines of subsection 2.5. Furthermore, for a sufficiently large value of G̃, it might
be possible to arrange for a≫ b at the outer surface, and thus avoid the problem that
prevented the solutions of subsection 2.5 from being in agreement with observation.
To study this possibility in detail, I shall now retrace the steps in subsection 2.5,
but assuming, now, that G̃ is sufficiently large, and the integration constant A, in
(210), is also sufficiently large, that as y increases, in the first bulk power law region,
(205), (206), and (207), with the integration constant, B, large compared to κ2/9, the
correction terms in (194) and (195), due to (357), first become significant long before
the 4
term, in the square root, becomes significant. This assumption will be satisfied,
if we find a solution in the classical region, with a≫ b at the outer surface.
With these assumptions, we find from (130), (194), (195), and (357), that the
relevant equations, away from the inner surface of the thick pipe, but still on the first
branch of the square root, where we take the upper sign in (194) and (195), are:
6c2 − 8 +
18b4a8
(358)
3c2 − 4
6c2 − 8 + G̃
18b4a8
36b5a8c
(359)
Here c = ḃ = db
, as defined just before (199), on page 88. These two equations
replace the equations (267) and (268) of subsection 2.5. Moreover, we are again seeking
solutions such that both a and b are large compared to κ2/9, at the outer surface of the
thick pipe, so, as explained at the start of subsection 2.5, on page 111, the boundary
conditions are now that both ȧ
, and ḃ
, are zero, at the outer surface of the thick
pipe. Thus c = 0 at the outer surface, and from (358), rewritten in its original form,
like (194), on page 87, we see that G̃
18b4a8
= 8, at the outer surface. The qualitative
difference from subsection 2.5, is that there we had b =
2a, b ≫ κ2/9, at the outer
surface, and here we have a new adjustable parameter, namely G̃
, related to the extra
fluxes, and we are going to try to choose a sufficiently large value of G̃
, that we find
a solution with a≫ b≫ κ2/9, at the outer surface.
Now the equations (358) and (359) have the family of solutions (206), (210), for
large values of the integration constants A and B, in the region c≫
, which means
b ≪ B, by (206), provided also that G̃
12b2a4
≪ 1, or in other words, by (210), provided
also that b
)0.9081
since, as noted above, for this class of solutions, 1
grows as b0.2024 with increasing b, hence 1
grows as b2.2024 with increasing b.
Let us now follow a solution of (358) and (359) in the class (206), (210), from small
b, in the direction of increasing b, and suppose that the G̃2 terms start to become
significant while the solution of (359), which is decoupled from (358) while these terms
are negligible, is still on the first branch of the square root. Thus the integration
constants A and B, in (206) and (210), must be such that κ2/9
)0.9081
is not
large compared to B. Then, in a similar manner to the situation in subsection 2.5,
when the G̃ terms first start to become significant, the solution of (359), in the (b, c)
plane, starts to peel off below the G̃ = 0 trajectory. Equation (359) then starts to
become coupled to equation (358), and we are looking for a solution such that the
trajectory, in the (b, c) plane, curves downwards and meets the line c = 0, at a finite
value of b, which will be b2 = b (y2), the value of b at the outer surface of the thick
pipe. At this point, a will take the value
In the limit c → 0, the other boundary condition, G̃2
18b4a8
= 8, implies that (359)
reduces to dc
= − 8
. This, in turn, reduces to dc
= − 8
, in the region of the boundary,
so that, in the region of the boundary, we have:
(360)
This replaces equation (269) of subsection 2.5, in the present context.
Following the method of subsection 2.5, we now define v ≡ 1
. The above equa-
tions then become:
6− 2
6c2 − 8 + G̃
 (361)
3c2 − 4
6c2 − 8 +
v2 (362)
Now, as noted shortly after (277), one way of studying a pair of equations of this type,
would be to take the ratio of (361) and (362). Then b cancels out, and we get a single
first order differential equation, that expresses dv
, as a function of v and c. However,
I shall follow the method of subsection 2.5. The boundary conditions, at b = b2, are
now that:
, c = 0 (363)
Near the boundary, we expand v in the small quantity
v = 12
1 + α
. Then from (360) and (361), we find that:
− α = 6− 2
6 + α (364)
which has the solutions:
α = −2, α = −6 (365)
We note that, analogously to the situation in subsection 2.5, the first of these is only
a solution, for the particular sign of the square root in (364), while the second is a
solution for both signs of the square root, since the square root vanishes for it. The
solutions can be developed to higher order in
, by substituting next into (362),
to fix the next term in c2, then after that into (361) again, to fix the next term in v,
and so on, in the same way as in subsection 2.5.
Furthermore, in a similar manner to subsection 2.5, (361) and (362) imply that:
6c2 − 8 + G̃
6c2 − 8 + G̃
6c2 − 8 + G̃
v2 − 1
4c2 +
(366)
Thus 6c2 − 8 + G̃2
v2 = 0 is a solution of (361) and (362). However the square root, R,
defined in (197), vanishes identically for this solution, so we cannot infer, from (198),
that 6c2 − 8 + G̃2
v2 = 0 is a solution of all three Einstein equations, and by analogy
with subsection 2.5, we would expect that it does not correspond to a solution of all
three Einstein equations, but is, rather, the generalization to the case where v 6= 0 of
the line c =
, which is the line in the (b, c) plane that actual solutions of the Einstein
equations, in the limit v = 0, osculate with as they switch from the first to the second
branch of the square root.
Moreover, the case α = −6, in (365), satisfies 6c2 − 8 + G̃2
v2 = 0, to the order
given, and is thus the c→ 0 limit of this particular solution of (361) and (362), and is
thus not expected to correspond to a solution of all three Einstein equations. We note
that this particular solution of (361) and (362) satisfies c ≤
, and thus can never
rise above the line c =
, in the (b, c) plane. Furthermore, when 6c2 − 8 + G̃2
v2 = 0,
(361) reduces to dv
, hence v = 12
, where, by (363), b2 is the integration
constant in (360). Hence c =
, which does, indeed, also solve (362).
Considering now, the case α = −2, in (365), and still following the method of
subsection 2.5, we see that 6c2−8+ G̃2
v2 ≃ 64
≃ 4c2 near the boundary, hence
the square root, R, is nonvanishing, as soon as we move away from the boundary, so,
by (198), this solution will correspond to a solution of all three Einstein equations.
Furthermore, for c ≥
, the right-hand side of (362) is ≤ the right-hand side of (204),
and as noted before (248), the right-hand side of (204) is ≤ 0 for all c ≥
, so the
right-hand side of (362) is ≤ 0 for all c ≥
, and the right-hand side of (362) is
certainly ≤ 0 for c ≤
such that the square root is real, so dc
is ≤ 0 for all c ≥ 0 and
b ≥ 0 such that the square root is real. Furthermore, dv
starts positive, specifically
at b = b2, hence v decreases, as b decreases downwards, away from b = b2,
hence, provided dv
never becomes negative, and the square root stays real, the square
root is bounded above by
6c ≃ 2.45c, hence, by (361), we have dv
1.10v
, and, by (366), we have d
6c2 − 8 + G̃2
24− 6
6c2 − 8 + G̃2
−9.30 c
6c2 − 8 + G̃2
v2, hence dv
never does become negative, and the square root does
stay real. Furthermore v ≤ 12
)1.10
, hence c ≥
, hence
6c2 − 8 + G̃2
6c2 − 8 + G̃2
≤ −9.30c
db ≤ −10.73
db (367)
Hence
6c2 − 8 + G̃
v2 ≥ 2.68
− arcsin
 (368)
With the bound v ≤ 12
)1.10
, this implies that 6c2−8 is positive for b
≤ 0.54, and is
greater than 13.45 for b
= 0.1, by which point G̃
v2 < 0.05. Thus this solution merges
into a solution of (204), as b continues to decrease, and for c large compared to
will follow a trajectory of the form (206), in the (b, c) plane, with B
a fixed number of
order 1, that will be the same for all solutions of this type. And for solutions of this
type, namely with α = −2 in (365), the constant of integration, b2, in (360), can be
identified as b2 = b (y2), the value of b at the outer surface of the thick pipe.
To estimate the integration constants B, in (206), and A, in (210), in terms of b2
and G̃, it is convenient to define ṽ ≡ G̃
v. The equations (361) and (362) then become:
6c2 − 8 + 8ṽ2
(369)
3c2 − 4− 4ṽ2
6c2 − 8 + 8ṽ2 (370)
The boundary conditions, at b = b2, are now that:
ṽ = 1, c = 0 (371)
Near the boundary, we have (360) and ṽ ≃ 1−2
≃ 1− c2
. Moreover, c increases
monotonically, and ṽ decreases monotonically, as b decreases downwards from b2. And
as b tends to zero, and c becomes large compared to 1, c tends to the form (206), where
B will be a fixed multiple of b2, that we now wish to estimate, and from (369), ṽ tends
to the form
ṽ ≃ V
)1.1010
)1.1010
c−0.5798 (372)
where the second form follows from (206), and the constant V is given, from (210),
and the relation ṽ = G̃
12b2a4
(373)
where A is the constant of integration that occurs in (210).
A simple estimate of the dependence of ṽ on c, with the required behaviour ṽ ≃ 1− c2
as c→ 0, and the power law behaviour (372) as c→ ∞, is
24− 8
1 + 0.4312c2
)−0.2899
(374)
which gives:
)1.1010
≃ 0.4312−0.2899 ≃ 1.2762 (375)
As a check on (374) and (375), we note that, on dividing (369) by (370), we have:
6c− 2
6c2 − 8 + 8ṽ2
3c2 − 4− 4ṽ2 − 2c
6c2 − 8 + 8ṽ2
(376)
And with the help of Maxima [291], we find that the solution of (376), that behaves as
ṽ ≃ 1− c2
near c = 0, has the Taylor expansion:
46080
− 3559c
10321920
15167c10
3715891200
+ · · · (377)
where . . . denotes terms of order c12. And using the graphical facility of Maxima, we see
that (377) is accurately approximated by its first four terms up to around c = 1.3, at
which point (377) is 0.8256, while (374) is 0.8533, and (377) is accurately approximated
by its first five terms up to around c = 2.0, at which point (377) is 0.6562, while (374)
is 0.7478. And (377) starts to curve rapidly downwards above around c = 2.0, and
would thus appear likely to depart from the true dependence of ṽ on c, starting at
around c = 2.0. Thus it seems likely that for large c, the estimate (374) of ṽ will be
around 15 to 20 percent too large, and the estimate (375), of V
)1.1010
, will also be
around 15 to 20 percent too large. So a better estimate of V
)1.1010
would perhaps
be around 1.1.
A simple estimate of the dependence of c on b, with the required behaviour (360)
as c→ 0, and b→ b2 from below, and the power law behaviour (206) for c≫
, is:
)1.8990
(378)
which also has the required property that c only depends on b, and the integration
constant b2, through the ratio
, as follows from (370), after substituting for ṽ as a
function of c, with a Taylor expansion that begins as in (377). The estimate (378)
leads to the estimate:
B ≃ 4
1.8990 b2 ≃ 2.0751b2 (379)
To check (378) and (379), we note that, from (370), we have:
4 + 4ṽ2 + 2x
6x2 − 8 + 8ṽ2 − 3x2
) (380)
where, in the integrand, ṽ is given as a function of x, by rewriting c as x, in the
dependence of ṽ on c as above, whose Taylor expansion begins as in (377). To calculate
0 fdx
by (380)
c for this
by (378)
(378)
c for this
by (381)
(381)
0.5 0.01561 0.9845 0.5130 1.0260 0.5032 1.0064
1.0 0.06170 0.9402 1.0997 1.0997 1.0209 1.0209
1.5 0.1329 0.8756 1.8156 1.2104 1.5577 1.0385
2.0 0.2177 0.8044 2.6746 1.3373 2.1081 1.0541
3.0 0.3821 0.6824 4.6576 1.5525 3.1991 1.0664
5.0 0.6286 0.5333 9.0170 1.8034 5.3261 1.0652
10.0 0.9866 0.3728 20.631 2.0631 10.621 1.0621
20.0 1.3504 0.2591 44.747 2.2374 21.220 1.0610
80.0 2.0801 0.1249 194.41 2.4301 84.842 1.0605
Table 1: The numerical dependence of c on b.
the integral in (380), I used the numerical integration facility of PARI/GP [274], with
the dependence of ṽ on c given by (377) for c ≤ 2, and by (374), multiplied by 0.6562
0.7478
0.8775, so as to obtain continuity at c = 2, for c ≥ 2. In this way, writing the integral
in the right hand side of (380) as
0 fdx, we find the entries in the second column of
Table 1. The entries in the third column are the values of b
which correspond by (380)
to the entries in the second column, and the entries in the fourth column are the values
of c which the estimate (378) gives, for the values of b
in the third column. The fifth
column gives the ratio of the estimated value of c calculated in the fourth column by
the estimate (378), to the original value of c in the first column.
From the form of the discrepancy factor in the fifth column, we would expect that
the estimate (378) could be improved by replacing the factor 4
, in (378), by
a factor of the form 4√
, which has the same limiting behaviour as b → b2
from below, and where
n ∼ 2.43 ∼
6. Thus, taking n = 6, we try an estimate:
c ≃ 1.633
)1.8990
(381)
The values of c given by the estimate (381) are listed in the sixth column of the table,
and from the discrepancy factor, in the seventh column of the table, we see that the
error now stays below 7 percent, and is actually slowly decreasing, as b
continues
to decrease below 0.5. I shall therefore use (381) as a reasonable estimate of the
ṽ from Taylor
series (377)
ṽ from
estimate (374)
(374)
0.5 0.9696 0.9696 0.9708 1.0012
1.0 0.8882 0.8882 0.9013 1.0147
1.5 0.7830 0.7818 0.8215 1.0492
2.0 0.6830 0.6562 0.7478 1.0949
3.0 0.5383 - 0.6315 1.1731
5.0 0.3919 - 0.4892 1.2483
10.0 0.2581 - 0.3336 1.2925
20.0 0.1719 - 0.2243 1.3048
80.0 0.07680 - 0.1006 1.3099
Table 2: The numerical dependence of ṽ on c.
dependence of c on b, in the presence of the extra fluxes. The corresponding estimate
of the integration constant B, in (206), is:
B ≃ 1.633
1.8990 b2 ≃ 1.2947b2 (382)
which now replaces the estimate (379).
Returning, now, to the dependence of ṽ on c, I used a standard fourth-order Runge-
Kutta method [292] to integrate (376) from c = 0.3, where ṽ is reliably given by (377)
as ṽ ≃ 0.9889, into the power law region, where c≫
. The same result was obtained
with a Runge-Kutta interval h = 0.01 as with h = 0.00001, even for c = 80. In fact, to
four significant digits, the same result was also obtained with h = 0.1, even for c = 80.
The results are shown in Table 2.
Thus the error of the estimate (374) stabilizes at about 31 percent in the power law
region, and the estimate (375) should be replaced by:
)1.1010
≃ 0.4312
−0.2899
≃ 0.9742 (383)
We next consider the dependence of b on y, and note, following the discussion shortly
after (277), in subsection 2.5, that the behaviour (360), for c near the boundary, implies
that near the boundary, y2− y ≃ b22
c, thus y does, indeed, tend to a finite
value, y2, at the boundary, even though
goes to ∞, right at the boundary. And
using the approximate relation (382), we find that near the outer boundary:
≃ 0.7724
1− 6.7050
y2 − y
(384)
For a first estimate of y2, we could simply use the form (207), with y0 = 0, all the way
from y1 to y2, and determine y2 as the point where this gives b = b2 ≃ 0.7724B, which
gives y2 ≃
0.7724
1.4436
0.3449
B ≃ 0.1631B. This underestimates y2 by a factor of order 1,
because, by (384), the curve of b (y), in the (y, b) plane, curves to the right as the outer
boundary is approached, so that b2 is not reached until a larger value of y than would
be indicated by (207) with y0 = 0.
For a better estimate of y2, a convenient interpolating function would be
b = 1.4436
)0.3449
(1 + αyβ)
γ (385)
with β > 0, and either α > 0 and βγ > 0.3449, or α < 0 and γ < 0. This agrees with
(207) for y ≪ 1
, and has a peak at y =
0.3449
α(βγ−0.3449)
, which we attempt to identify
with y2. Requiring agreement with (384) for b (y2) leads to the requirement that:
1.4436
0.3449
0.3449
(βγ − 0.3449)
βγ−0.3449
β = 0.7724 (βγ)
(386)
This is written for the α > 0 case, and for the α < 0 case should be rewritten in the
equivalent form with the contents of each of the three pairs of parentheses multiplied
by −1. And requiring agreement with (384) for 1
, evaluated at y = y2, leads to the
requirement that:
βγ − 0.3449
0.3449
= 112.73γ (387)
This is also written for the α > 0 case, and for the α < 0 case should be rewritten in
the equivalent form with the contents of each of the two pairs of parentheses multiplied
by −1, and the right hand side also multiplied by −1. Eliminating αBβ between (386)
and (387), we find:
βγ − 0.3449
)γ+0.1725
1.0059
β0.1725
(388)
Trying first β = 1, there is no solution in the α > 0 case, but there is a solution
with γ ≃ −0.1672 in the α < 0 case. We note that the improved estimate, (381),
of the dependence of c on b, shows that the power law behaviour remains a good
approximation until b is quite close to b2. For small
, the correction to the power
β γ αBβ y2
1 −0.1672 −2.3996 0.2807
2 −0.6666 −3.1744 0.2545
2.5 2.4194 2.0098 0.2462
4 0.1771 340.47 0.2298
Table 3: Parameters for the interpolating function (385) for b as a function of y.
law behaviour, in (381), is by a term of relative size
, which is ∼
)2.0694
in the
power law region, which suggests that β = 2 might be a good choice in (385). However
with β = 2, there still appears to be no solution in the α > 0 case, while there is a
solution with γ ≃ −0.6666 in the α < 0 case. Trying β = 2.5, there is a solution with
γ ≃ 2.4194 in the α > 0 case, but apparently no solution in the α < 0 case. And
trying β = 4, there is a solution with γ ≃ 0.1771 in the α > 0 case, and apparently
no solution, again, in the α < 0 case. These example solutions, and the corresponding
values of αBβ and y2
, are listed in Table 3.
We see that, notwithstanding the substantial differences between the parameters of
the interpolating function, for the different choices of β, the corresponding values of
only differ by around 20 percent, and are around 1.4 to 1.7 times larger than the
value 0.1631 given by the uncorrected power law (207). They show a trend towards
the uncorrected power law value with increasing β, corresponding to a later and more
rapid onset of the corrrections to the power law.
We can also obtain an approximate value of y2 by integrating the approximate
formula (381). From (381) and (382) we obtain:
∫ 0.7724
x1.8990dx
0.7724
≃ y2 − y1
(389)
But b1 ∼ κ2/9, which by assumption is small compared to B, so we can extend the
lower limit of the integral in the left hand side of (389) to zero, and choosing, as usual,
the integration constant y0 in (207) to be zero, we have y1 ≪ κ2/9 ≪ B, so we can drop
the y1 term in the right hand side of (389). Then by use of the numerical integration
facility of PARI-GP [274], plus an analytic approximation for the contribution from
the region close to the upper limit, the integral in the left hand side of (389) is found
to be ≃ 0.2536. And comparing with the estimates of y2
as given in Table 3, for the
different choices of β in the interpolating function (385), we see that the best agreement
is obtained for the choice β = 2, as expected from the discussion following (388).
2.7.2 Newton’s constant and the cosmological constant in the presence of
the extra fluxes
Turning now to fitting the observed values of Newton’s constant and the cosmological
constant, we again follow the method used in subsection 2.5.1, on page 115. The term,
in the Einstein action term in (25), that produces the Einstein action, (10), in four
dimensions, is again given by (278), where b (y) is now given approximately by (385),
with β preferably chosen as 2, and γ and α as given by the row corresponding to β = 2
in the above table, and a, as a function of b, is given, as a first approximation, in
terms of the approximate dependence of ṽ on c, in (374), the approximate dependence
of c on b, in (381), and the relation ṽ = G̃
12b2a4
. The worst approximation here is the
estimate (374) of the dependence of ṽ on c, which has a percentage error that stabilizes
at around 31 percent in the power law region, as found above. We then have:
dya2b6 =
a2b6 =
∫ 0.7724B
(390)
We note that since ṽ occurs in (390) only through its square root, the contribution to
the error percentage resulting from the use of (374) will be roughly halved, to not more
than around 16 percent. And in the power law region, where by (372), ṽ ∼ c−0.5798, the
denominator, in the last integral in (390), is ∼ c0.7101, so since (381) overestimates c
by not more than about 7 percent, and (374) will overestimate
ṽ, as a function of c,
by not more than about 16 percent, the use of (381) and (374), in (390), is expected to
give a result that will be smaller than the correct result, but by not more than about
16+0.71× 7 ≃ 21 percent. Furthermore, since c ∼ b−1.8990 in the power law region, by
(206), the integrand, in the last integral in (390), is ∼ b6.3485 in the power law region,
and goes to infinity as (0.7724B − b)−
2 at the upper limit, due to the behaviour (360)
of c, so the last integral in (390) is substantially dominated by the contribution from
the region near the upper limit, where (381) and (374) are accurate, so the error is in
fact expected to be substantially smaller than 21 percent.
Inserting the approximate expressions (381) and (374), we find:
dya2b6 ≃ B6
∫ 0.7724
x6.8990dx
1 + 0.4312
x3.7980
0.7724
))0.1450
0.7724
(391)
Now as before, b1
∼ κ2/9
, so we can set the lower limit of the integral in the right hand
side of (391) to zero. Then using again the numerical integration facility of PARI-GP,
plus an analytic approximation for the contribution from the region near the upper
limit of the integration domain in the integral in the right hand side of (391), we find:
dya2b6 ≃ 0.03896B6
(392)
And, as explained above, the coefficient 0.03896 is expected to be smaller than the
correct value, but the percentage error is expected to be substantially smaller than 21
percent.
Now from (373) and (383), we have:
≃ 0.9742κ
)1.1010
(393)
Hence from (392), and (241), on page 104, we have:
dya2b6 ≃ 0.03845κ
)5.4495
≃ 0.03845κ
9 A21
)1.3102τ+6.4653
(394)
which replaces (280), on page 117, and (308), on page 127, for the present situation,
where the outer boundary is controlled by the extra fluxes. We see that, as found in
subsections 2.5.1 and 2.6.1, there is no ADD effect unless τ > −4.9346.
Continuing to follow subsections 2.5.1 and 2.6.1, we now find that when the compact
six-manifold is a smooth compact quotient of CH3, the Einstein action term, in the
four-dimensional effective action, for the solutions considered in the present subsection,
will be equal to:
0.3974
)5.4495
−g̃g̃µνRµτντ (g̃) (395)
I shall now consider the case where τ = −0.7753, which corresponds to the clas-
sical power law (210), on page 90. Then defining the rescaled metric ḡµν by ḡµν =
(de Sitter radius)
g̃µν , as in subsections 2.5.1 and 2.6.1, so as to measure distances
in ordinary units rather than in units of the de Sitter radius, we find from (242), on
page 104, that the Einstein action term in the four-dimensional effective action, for
the solutions considered in the present subsection, will for τ = −0.7753 and smooth
compact quotients of CH3 be equal to
− 0.5808
)5.4495
0.7416
−ḡḡµνRµτντ (ḡ) (396)
Thus, comparing with (10), we find that for τ = −0.7753 and smooth compact quotients
of CH3:
≃ 29.19 1
)5.4495
0.7416
(397)
This is the form taken by the ADD mechanism [3, 5], for the solutions considered in
the present subsection with τ = −0.7753. Thus for τ = −0.7753 and smooth compact
quotients of CH3:
≃ 0.5384
|χ (M6)|0.1361
0.1835
(398)
Considering, now, the case of TeV-scale gravity, I shall again consider the case where
9 = 0.2217 TeV, so that κ2/9 = 8.899 × 10−19 metres, and the Giudice, Rattazzi,
and Wells [11] gravitational mass MD, for D = 11, is equal to 1 TeV. We then find,
from (12), that for τ = −0.7753 and smooth compact quotients of CH3:
7.499× 105
|χ (M6)|0.1361
κ2/9 ≃
6.673× 10−13 metres
|χ (M6)|0.1361
(399)
On the other hand, for τ = −0.7753, the integration constant A in (210), on page 90, is
from (242), on page 104, fixed directly in terms of |χ (M6)| and the observed de Sitter
radius (22), on page 15, and given for smooth compact quotients of CH3 by (319), on
page 130. Hence from (393) we find, for τ = −0.7753 and smooth compact quotients
of CH3, that:
G̃ ≃ 3.53× 10
63metres6
|χ (M6)|0.3670
≃ 7.11× 10
|χ (M6)|0.3670
4.39× 1028κ2/9
|χ (M6)|0.3670
(400)
where G̃ is defined in (352). This is the large constant of integration, not constrained
by the field equations or boundary conditions, that is built into the structure of the
universe, to make it into the stiff, strong structure that we observe, for the solutions
considered in the present subsection with τ = −0.7753, in the case of TeV-scale gravity.
In a similar way to the situation with τ , Ã
, and B
in subsection 2.6.1, it will
be possible, by decreasing τ below −0.7753, to decrease G̃
at a cost of increasing B
until as τ approaches the values near −3 in (286) and (287), on page 118, it will no
longer be a good approximation to neglect the term 4
in the square root in comparison
to the term G̃
18b6a8
, and the solutions considered in this subsection will tend as G̃ → 0
to those studied in subsection 2.5.1, on page 115.
More generally, from (242), (22), (398), and (393), we find that for τ = −0.7753
and smooth compact quotients of CH3, with a general value of κ2/9:
3.90× 10245
|χ (M6)|0.3670
)2.2020
(401)
Thus the required value of G̃
is minimized by choosing κ2/9 to be as large as possible,
which means TeV-scale gravity, provided this is consistent with the precision tests of
Newton’s law down to sub-millimetre distances [32]. To check that this requirement is
satisfied, we now determine the values of b, y, and a, at the outer surface of the thick
pipe, for the solutions considered in the present subsection with τ = −0.7753, in the
case of TeV-scale gravity, with κ−
9 = 0.2217 TeV.
From (399) and (382), we find that b2, the value of b at the outer surface of the
thick pipe, is given, for τ = −0.7753 and smooth compact quotients of CH3, by:
5.15× 10−13 metres
|χ (M6)|0.1361
(402)
And since, from above, y2, the value of y at the outer surface of the thick pipe, which
is also the “radius” of the thick pipe, is approximately given by y2 ≃ 0.254B, we
find, from (399), that for τ = −0.7753 and smooth compact quotients of CH3, y2 is
approximately given, for TeV-scale gravity, by:
y2 ≃ 0.254B ≃
1.70× 10−13 metres
|χ (M6)|0.1361
(403)
Furthermore, from (400), (402), the relation ṽ = G̃
12b2a4
, and the boundary condition
(371) on ṽ at the outer boundary, where b = b2, we find that for τ = −0.7753 and
smooth compact quotients ofCH3, a2 = a (y2), the value of a (y) at the outer boundary,
is given by:
12b22
5.77× 1021 metres
|χ (M6)|0.0237
(404)
We note that, since |χ (M6)| ≥ 1, this is large compared to b2, as assumed near the
beginning of this subsection. Furthermore, since a1 = a (y1) is the de Sitter radius (22),
and |χ (M6)| is bounded above by around 7 × 104, the ratio a1
is bounded above by
around 3.4×104. Thus 0.2 millimetres on the inner surface of the thick pipe corresponds
on the outer surface to a distance no shorter than around 6 nanometres, so the four
dimensional effective field theory description is certainly valid for distances down to
0.2 millimetres, and the realization of TeV-scale gravity considered in this subsection is
for τ = −0.7753 consistent with the precision tests of Newton’s law at sub-millimetre
distances.
Turning now to the flux quantization condition, (345), we find, from the relation
ṽ = G̃
12b2a4
, and the approximate relation (382) between b2 = b (y2) and B, that
dya (y)
−4 ≃ 12
∫ 0.7724B
b2ṽ (405)
Inserting the approximate dependence of ṽ on c, in (374), and the approximate depen-
dence of c on b, in (381), we find:
dya (y)
∫ 0.7724
x3.8990dx
0.7724
1 + 0.4312
x3.7980
0.7724
))0.2899 (406)
For small b, or equivalently, for large c, the integrand in the right-hand side of (405)
behaves as b
c1.5798
, and thus as b5, so the integral is dominated by the contribution from
the region near the upper limit. The estimate (374) of the dependence of ṽ on c is
accurate near the upper limit, and becomes too large by about 31 percent at large c,
and the estimate (381) of the dependence of c on b is accurate near the upper limit,
and becomes around 6 percent too large at small b, so the integrand in the right-hand
side of (406) is accurate near the upper limit, and too large by around 22 percent near
the lower limit. Thus we expect (406) to give a result that is too large, but by a lot less
than 22 percent. We can again set the lower limit to zero, since b1
∼ κ2/9
, and using
again the numerical integration facility of PARI-GP, plus an analytic approximation
near the upper limit, we find:
dya (y)
−4 ≃ 1.0907B
(407)
Thus from the flux quantization condition (345), the quantity that is required to be an
integer, for each three-cycle, Z, of the compact six-manifold, is approximately:
0.2212
dzAdzBdzCGABC (z) (408)
Now comparing with the definition (352) of the constant G̃, we see that G̃ cancels out of
(408), which is thus independent of the overall normalization ofGABC (z). Thus the flux
quantization condition (345) does not constrain the integration constant A in (210),
the de Sitter radius (22), or the effective cosmological constant in four dimensions,
(20). Furthermore, we recall that GABC (z) is a linear combination, with position-
independent coefficients, of the B3 linearly independent Hodge - de Rham harmonic
three-forms on the compact six-manifold M6, where B3 is the third Betti number of
M6, that has been assumed to satisfy the conditions (347) and (348), which constitute
at most 20 + 20 linearly independent constraints on the B3 independent coefficients in
GABC (z).
We can always choose a linearly independent set of B3 Hodge - de Rham harmonic
three-forms g
ABC (z), 1 ≤ i ≤ B3, and a set of B3 three-cycles Z(j) of M6, 1 ≤ j ≤ B3,
linearly independent in the sense of homology, such that
dzAdzBdzCg
ABC (z) =
(j). Choosing a basis of harmonic three-forms and a set of B3 three-cycles that
satisfy this condition, the requirement that (408) be an integer, for each three-cycle Z(j),
1 ≤ j ≤ B3, implies that the (B3 − 1) independent ratios of the coefficients in GABC (z)
are rational numbers. The overall normalization of the coefficients, which cancels out of
(408), is fixed by (400) for TeV-scale gravity, and by (401) in general, together with the
definition (352) of G̃. If we now define ρj ≡ 1G̃
dzAdzBdzCGABC (z), 1 ≤ j ≤ B3,
the requirement that (408) be an integer for all three-cycles Z of M6 reduces to the
requirement that 0.2212B
ρj be an integer for all 1 ≤ j ≤ B3.
Now the value of B
has been assumed to be fixed by the boundary condition at
the inner surface of the thick pipe, with its actual value determined by the Casimir
energy densities on and near the inner surface of the thick pipe, so B
would be
overconstrained if the flux quantization conditions significantly restricted its value.
However for TeV-scale gravity, (399) implies that the value of B
is around 1016, pro-
vided |χ (M6)| is not too large, so provided none of the nonvanishing ρj are too small
in magnitude, and the nonvanishing
are expressible as ratios of sufficiently small
integers, an alteration of a (y) by a tiny percentage in the region near the outer sur-
face, where the alteration would have the greatest effect on the integral (405), would
be sufficient to satisfy all the flux quantization conditions.
Furthermore we are free to choose the independent ratios of the ρj , and thus to set
them equal to ratios of small nonvanishing integers, in which case it seems plausible
that the magnitudes of the ρj would generally lie more or less within the range
. Thus provided |χ (M6)| and B3 are not too large, it seems plausible, at least
for the case of TeV-scale gravity, that the flux quantization conditions, (345), will not
significantly restrict the solutions considered in the present subsection.
We note that, notwithstanding the large value (400) of G̃ in the case of TeV-scale
gravity, and its large value (401) in general, the extra fluxes of the four-form field
strength of the three form gauge field considered in the present subsection, which
wrap three-cycles of the compact six-manifold times the radial dimension of the thick
pipe, never have a large enough field strength that we would expect them to produce
quantum effects. To estimate whether we would expect the extra fluxes to produce
quantum effects, we note that we expect quantum gravitational effects when the Ricci
scalar has magnitude ∼ κ− 49 or larger. Hence from the supergravity action (25), we
would expect the four-form field strength GIJKL to produce quantum effects when
GIMGJNGKOGLPGIJKLGMNOP has magnitude ∼ κ−
9 or larger. And, noting that
there are no cross terms in GIMGJNGKOGLPGIJKLGMNOP between the extra fluxes
and the standard Witten fluxes that follow for smooth compact quotients of CH3
from Witten’s topological constraint, as studied in subsections 2.3.7 and 2.3.8, we find,
from (351), that in the approxiation of dropping all but the leading harmonic, the
contribution to GIMGJNGKOGLPGIJKLGMNOP from the extra fluxes is given by
GQRGKNGLOGMPGQKLMGRNOP =
G̃2 = 576
(409)
where the relation ṽ = G̃
12b2a4
was used. Thus since ṽ = 1 at the outer surface of the
thick pipe, by (371), and b ∼ B at the outer surface of the thick pipe, which for TeV-
scale gravity is ≫ κ2/9 by (399), unless |χ (M6)| is extremely large, which seems very
unlikely since it would require a correspondingly small value of b1
, by (103), we see
that for TeV-scale gravity the extra fluxes are not large enough at the outer surface of
the thick pipe that we would expect them to cause quantum effects there.
Furthermore, from (210), on page 90, 1
behaves as b0.2024 in the bulk power law
region, and thus decreases with decreasing b, and from the form of the solutions studied
above, neither ṽ nor b changes significantly in order of magnitude between the power
law region and the outer surface of the thick pipe, hence for TeV-scale gravity (409) is
small in magnitude compared to κ−
9 throughout the whole thick pipe. Thus for TeV-
scale gravity, we do not expect the extra fluxes considered in the present subsection
to produce any significant quantum effects at all, and away from the inner surface of
the thick pipe, the solutions studied in the present subsection are entirely classical in
character.
We note, furthermore, that even though G̃, and the integration constant, A, in
(210), cancel out of the reduced Einstein equations (369) and (370) and boundary
conditions (371), and also cancel out of the contribution of the extra fluxes to
GIMGJNGKOGLPGIJKLGMNOP , they are nevertheless physically significant. For first
of all, if G̃ and A had not been sufficiently large, it would not have been possible to
neglect the term 4
in the square root in (194) and (195), in comparison with the new
term in 2
κ2t(3) coming from (357), as explained between (357) and (358), and it would
not then have been possible to eliminate G̃ from the Einstein equations by defining
ṽ = G̃
12b2a4
. And secondly, from the metric ansatz (94), and (210), the observed de Sitter
radius (22), as estimated from observations of type Ia supernovae [293, 294, 295], and
significantly bounded below by a great variety of astronomical observations, as well
as by the approximate flatness of the everyday world, is equal to A
)0.7753
, where
b1 = b (y1) is expected, by (103), to be ∼ κ2/9. It is the large value of A, which for the
solutions considered in the present subsection results from the large value of G̃, that
results in the existence of a large and approximately flat platform at the inner surface
of the thick pipe, on which the interesting processes of intermediate range astronomy,
and the everyday world, can take place.
The value of G̃ is not constrained by the field equations or the boundary conditions,
since the relevant field equation, (342), is satisfied both in the bulk and on the orbifold
fixed-point hyperplanes, in the upstairs picture, in consequence of the Hořava-Witten
orbifold conditions, as summarized after (25), which imply that the components GyUVW
are even under reflections in the orbifold hyperplanes. It seems that the existence of the
arbitrary constant G̃, defined in (352), in compactifications of Hořava-Witten theory
of the type studied in the present paper, is implicit in the field content of supergravity
in eleven dimensions [38, 14], and the boundary conditions or orbifold conditions of
Hořava-Witten theory, as summarized after (25). The question of how G̃ came to have
the large value required to fit the observed small value of the cosmological constant,
and the related question of whether G̃ has any effects on the dynamics of the early
universe, other than preventing the occurrence of a large cosmological constant, in
models of this type, will not be considered in the present paper.
Finally we should check whether the solutions considered in the present subsection
are consistent with the precision tests of Newton’s law down to submillimetre distances,
and the very high precision tests of Newton’s law over solar system distances, as carried
out for the solutions of subsection 2.6 in subsection 2.6.1, starting shortly after equation
(321), on page 132. However, it does not seem very likely that the constraints from
these tests will be more stringent for the solutions considered in the present subsection
than for the solutions of subsection 2.6, and this will not be considered in detail in the
present paper.
The fact that the value of G̃ does seem to be quantized seems to suggest that
if the metric ansatz (94) is generalized to a cosmological ansatz of the form ds211 =
−a (y)2 dt2 + ã (y, t)2 gijdxidxj + b̃ (y, t)2 hABdxAdxB + c̃ (y, t)2 dy2, which is consistent
with (94) if the metric on the four-dimensional de Sitter space, in (94), is taken in
Friedmann-Robertson-Walker form, a parameter related to G̃ might evolve with time,
as in quintessence models [33]. This could perhaps be investigated by studying cosmo-
logical solutions that are small perturbations of the de Sitter solutions studied in this
section.
3 Smooth compact quotients of CH
, H6, H3 and S3
For the compactifications of Hořava-Witten theory considered in the present paper,
the compact six-manifold, M6, is a smooth compact quotient of either the symmetric
space CH3, or the symmetric space H6, by a discrete subgroup of the isometry group
of the symmetric space, and for the solutions considered in subsection 2.6, on page
120, the three observed spatial dimensions are also compactified to a smooth compact
quotient of either the symmetric space H3, or the symmetric space S3, by a discrete
subgroup of the symmetric space. I shall first consider smooth compact quotients of
the non-compact symmetric spaces CH3, H6, and H3, then briefly consider smooth
compact quotients of S3, at the end of this section.
Let G be the identity component, or in other words, the connected component that
contains the identity, of either SU(n,1), for n ≥ 1, or SO(n,1), for n ≥ 2, and let K be
the maximal compact subgroup of G, which is SU (n)×U (1) for SU(n,1), and SO(n),
for SO(n,1). Then the non-compact symmetric space, S ≡ G/K, is CHn for SU(n,1),
and Hn for SO(n,1). I shall assume that the metric of CHn is normalized such that
the Riemann tensor is given by (72), in complex coordinates, and that the metric of
Hn is normalized such that the Riemann tensor is given by Rµνστ = gµσgντ − gµτgνσ,
consistent with the choice made between (188) and (189), and between (99) and (100),
so that the sectional curvature (84) of Hn is equal to −1, which is the conventional
value.
We choose a point of S, called O, to be the origin of S. For example, for CHn,
we could choose O to be the origin of the coordinates used for CHn, in subsection 2.2.
For any subgroup, H , of G, let C (H) denote the set of all the set of all the images of
O, by the action of elements of H . A discrete subgroup, Γ, of G, is a subgroup of G
such that there is a real number ρ > 0, such that for all members x of C (Γ) different
from O, the geodesic distance from O to x is ≥ ρ. For any discrete subgroup, Γ, of
G, and any member, x, of C (Γ), the Wigner-Seitz cell, or Voronoi cell, W (Γ, x), is the
set of all points of S, that are closer to x, than to any other member of C (Γ). The
fundamental domain of the quotient S/Γ is W (Γ, O). Γ is called a lattice in G, if
W (Γ, O) has finite volume. When Γ is a lattice in G, the set C (Γ), of all the images
of O in S, looks like a hyperbolic analogue of a crystal lattice.
G always has an infinite family of lattices called “arithmetic lattices”, whose exis-
tence was demonstrated by Borel and Harish-Chandra [34]. A very helpful introduction
to arithmetic lattices has been provided by Morris [296]. Some examples of arithmetic
lattices in G are considered in subsection 3.1, on page 167. For SO(n,1), SU(2,1),
and SU(3,1), there also exist additional lattices called “non-arithmetic lattices”. Non-
arithmetic lattices in SO(n,1), for n ≤ 5, were constructed by Makarov and Vinberg
[297, 298], and non-arithmetic lattices in SO(n,1), for all n, were constructed by Gro-
mov and Piatetski-Shapiro [35]. The construction of Gromov and Piatetski-Shapiro
involves cutting two different quotients of S into two parts along totally geodesic n-
dimensional submanifolds, and smoothly joining together one part from each of the
two different quotients. There is no analogous construction for SU(n,1) for n ≥ 2,
because for n ≥ 2, CHn has no totally geodesic (2n− 1)-dimensional submanifolds
[121]. Non-arithmetic lattices in SU(2,1) were constructed by Mostow [36], and non-
arithmetic lattices in SU(3,1) were constructed by Deligne and Mostow [37]. To the
best of my knowledge, it is not yet known whether or not any non-arithmetic lattices
exist for SU(n,1), n ≥ 4.
For the models considered in the present paper, I assume that the quotient S/Γ
is a smooth manifold, not an orbifold, so Γ is required to act on S without fixed
points, or in other words, no non-trivial element of Γ is allowed to leave any point of
S invariant. The fact that S is the quotient of G, by its maximal compact subgroup,
K, implies that a necessary condition for Γ to act without fixed points, is that Γ must
have no torsion, in the sense of discrete group theory, or in other words, Γ must not
contain any element g 6= 1, such that gn = 1, for some finite n. For the finite group
generated by such a g is a compact subgroup of G, and every compact subgroup of
G is contained in a maximal compact subgroup, and all maximal compact subgroups
of G are conjugate. Furthermore, K is the subgroup of G that leaves O invariant,
and the conjugate hKh−1 of K, where h is a fixed element of G, leaves the point hO
invariant. Conversely, the requirement that Γ have no torsion is also sufficient to ensure
that Γ acts on S without fixed points. For suppose an element g 6= 1 of Γ leaves a
point x of S invariant. Then since Γ is an isometry, and maps members of C (Γ) to
members of C (Γ), g must permute the members of C (Γ), at any given fixed distance
from x, among themselves. In particular, g must permute the nearest neighbours of
x, in C (Γ), amongst themselves. But the discreteness of Γ implies that the number
of nearest neighbours of x, in C (Γ), is finite. Hence g is an element of a finite group,
hence gn = 1, for some finite n.
Now let Γ be a lattice in G, that acts without fixed points on S, so that the quotient
S/Γ is a smooth manifold, of finite volume. Let d denote the real dimension of S,
which is 2n for SU(n,1), and n for SO(n,1). Then for d ≥ 3, Mostow’s rigidity theorem
[299, 300, 10, 301, 302] implies that the locally symmetric space S/Γ is completely
determined, up to isometry, by its fundamental group, which is Γ. This result is not
true for d = 2, since smooth compact quotients of CH1, which differs from H2 only in
the normalization of its metric, in the conventions adopted here, have shape moduli, as
is well known in superstring theory. An orientable smooth compact quotient of CH1 of
genus g ≥ 2, which is topologically equivalent to a sphere with g handles, has a moduli
space of dimension 6g − 6. The moduli are called Teichm”uller parameters, and are
the minimum number of parameters needed to characterize conformally inequivalent
closed Riemann surfaces. They correspond to the positions and radii of six circles in
the complex plane, which are identified in pairs to produce the closed surface, less six
parameters that relate conformally equivalent surfaces.
Mostow’s rigidity theorem implies, in particular, that for d ≥ 3, the volume,
V (S/Γ), is a topological invariant. For d even, V (S/Γ) is a fixed multiple of the
Euler number, given for d = 6 by (99) for smooth compact quotients of CH3, and by
(100) for smooth compact quotients of H6, but for d odd, the Euler number is zero,
and, at least for d = 3, there is no corresponding restriction on the possible values of
V (S/Γ).
If there is a finite upper bound, on the geodesic distance between pairs of points
in the fundamental domain W (Γ, O), then the quotient S/Γ is compact, while if, for
any given finite distance, there exist pairs of points, in the fundamental domain, such
that the geodesic distance between them is greater than that given distance, then the
quotient is non-compact. In the coordinate system used for CHn, in subsection 2.2,
a quotient of CHn is compact, if all points in the closure of its fundamental domain
have zrzr̄ < 1, and non-compact if the closure of its fundamental domain has one or
more vertices on the sphere zrzr̄ = 1. If a quotient is non-compact, then the non-
compactness is associated with a finite number of tubular regions, called cusps, which
extend out to infinite distances, but become narrow so rapidly, that their contribution
to the total volume is finite.
Inspection of the list in section (14.4) of [37] shows that the non-arithmetic quotients
of CH3 found by Deligne and Mostow, which correspond to the lattice in SU (3, 1)
denoted 753333 in their notation, are not compact. This same lattice is also the only
non-arithmetic lattice in SU (3, 1) listed in the Appendix in [303], where it occurs as No.
66 in the list. Thus to the best of my knowledge, it is at present not known whether or
not any compact non-arithmetic quotients of CH3 exist. However, the non-arithmetic
quotients of CH2 found by Mostow in [36] are compact.
For the models considered in the present paper, I assume that the quotient is
compact. However, smooth non-compact finite volume quotients of H3 are important,
because by a construction of Thurston [301, 304, 305, 276], there exist infinite sequences
of smooth compact quotients of H3, all with distinct topology, whose volumes converge
to the volumes of smooth non-compact finite volume quotients of H3. This cannot
happen in any dimension larger than 3, because by a theorem ofWang [277], the number
of topologically distinct smooth finite volume quotients of a non-compact symmetric
space of dimension ≥ 4, whose volume V (S/Γ) is less than a given volume, is finite.
Moreover, Borel [306] demonstrated that the number of topologically distinct smooth
compact arithmetic quotients of H3, whose volume is less than a given volume, is
finite, so all but a finite number of the smooth compact quotients of H3 resulting from
Thurston’s construction, whose volume is less than a given volume, are non-arithmetic.
Each cusp of a non-compact finite volume quotient of H3 is topologically equivalent to
the Cartesian product of a two-torus and the infinite half-line.
Thurston’s construction makes use of a method of modifying the topology of a
three-manifold M3, called Dehn surgery [307, 308, 309]. A Dehn surgery consists of
removing a tubular neighbourhood N of an S1 embedded in the manifold, then putting
it back, in a twisted fashion. The surface of N is a two-torus, and twists can be
introduced in two independent senses. We choose two oriented simple closed curves
m and l, called the meridian and the longitude, embedded in the common boundary
torus of N and its complement, that generate the fundamental group of that torus.
When M3 is S3, l is chosen such that it bounds a surface in the complement of N , and
m is chosen such that it crosses l exactly once. This gives any oriented simple closed
curve c on that torus two coordinates p and q, which correspond to the net number of
times c crosses m and l respectively. These coordinates depend only on the homotopy
class of c. The Dehn surgery with slope u = p
, where p and q are coprime integers,
then corresponds to gluing back N by means of a homeomorphism of its two-torus
boundary to the two-torus boundary of its complement, such that the meridian curve
of the boundary of N maps to a (p, q) curve in the boundary of its complement. By a
theorem of Lickorish [310] and Wallace [311], every closed, orientable, connected three-
manifold can be obtained from S3 by Dehn surgery around finitely many copies of S1
embedded disjointly in S3.
Now let M3 a smooth non-compact finite volume quotient of H3 with n cusps,
where n ≥ 1. Then because each cusp of M3 is topologically equivalent to the
Cartesian product of a two-torus and the infinite half-line, M3 is topologically equiv-
alent to the interior of a compact three-manifold with n connected boundary compo-
nents, each of which is topologically equivalent to a two-torus. We choose a merid-
ian and longitude for each boundary torus, as in the case of Dehn surgery. Let
M3(u1, u2, . . . , un) denote the manifold obtained from M3 by filling in each bound-
ary two-torus with a solid torus using the slopes ui = pi/qi, where each pair pi and qi
are coprime integers. This is called Dehn filling. Thurston’s hyperbolic Dehn surgery
theorem then states that M3(u1, u2, . . . , un) is topologically equivalent to a smooth
compact quotient H3/Γ (u1, u2, . . . , un) of H
3, provided a finite set Ei of slopes is
avoided for each i. Furthermore, V (H3/Γ (u1, u2, . . . , un)) < V (M3), and the volumes
V (H3/Γ (u1, u2, . . . , un)) converge to V (M3) as all p2i + q2i → ∞, pi 6= 0, qi 6= 0.
It is also known that only a finite number of topologically distinct smooth compact
quotients of H3 with any given volume exist [304].
Many smooth non-compact finite volume quotients of H3 have been discovered by
studying the complement of disjoint tubular neighbourhoods of finitely many copies
of S1 embedded disjointly in S3, to see if it can be constructed by gluing together a
small number of hyperbolic tetrahedra, some of whose vertices stretch out to infinity
as parts of cusps. A hyperbolic polyhedron, one or more of whose vertices stretches out
to infinity as part of a cusp, is called an ideal hyperbolic polyhedron. This method was
originally applied by Thurston to show that the complement of the figure of eight knot
was hyperbolic, by constructing it by gluing together two ideal hyperbolic tetrahedra.
It had earlier been shown to be hyperbolic by Riley, and by Jorgensen, using other
methods.
Weeks’s computer program SnapPea [312], which can perform Dehn surgeries au-
tomatically, includes a census of smooth non-compact finite volume quotients of H3
constructed by gluing together up to seven hyperbolic tetrahedra. The smooth non-
compact finite volume quotient ofH3 that is topologically equivalent to the complement
of the figure of eight knot is designated m004 in SnapPea, and has volume 2.02988 . . ..
This has been shown by Cao and Meyerhoff [313] to be the smallest possible volume
of an orientable cusped hyperbolic three-manifold. There is one other known smooth
non-compact finite volume quotient of H3 with this volume, which is designated m003
in SnapPea, and can be obtained by a (5, 1) Dehn filling on the complement of the
Whitehead link. The Whitehead link is a disjoint embedding of two copies of S1 in S3,
such that neither S1 is knotted by itself, but the two copies of S1 are linked such that
one S1 has a loose twist to resemble a figure of eight, and the other S1 links both loops
of the figure of eight.
The smooth compact quotient of H3 of smallest known volume is called the Weeks
manifold or the Fomenko-Matveev-Weeks manifold [314, 315], and can be obtained by
a (5,2) Dehn filling on m003 or by a (3,−1) Dehn filling on m003, and has volume
0.9427 . . .. The smooth compact quotient of H3 of second smallest known volume is
called the Thurston manifold, and can be obtained by a (−2, 3) Dehn filling on m003,
and has volume 0.9814 . . ..
By Thurston’s hyperbolic Dehn surgery theorem, there are already an infinite num-
ber of topologically distinct smooth compact quotients of H3 with volume less than
the volume 2.02988 . . . of m003 and m004, while as noted above, when the dimension
d of S is ≥ 4, the number of topologically distinct smooth finite volume quotients of
S, whose volume V (S/Γ) is less than a given volume, is finite. Let ρS (v) denote the
number of topologically distinct smooth finite volume quotients of S, whose volume
V (S/Γ) is less than v. Then Gelander [316] has proved that when the dimension d of
S is ≥ 4, there is a constant c, depending on S, such that
log ρS (v) ≤ cv log v (410)
for all v > 0. And for Hn, n ≥ 4, Burger, Gelander, Lubotzky, and Mozes (BGLM)
[317] have proved that there exist constants cn > bn > 0 and vn > 0, such that
bnv log v ≤ log ρHn (v) ≤ cnv log v (411)
whenever v > vn.
Thus the number of topologically distinct smooth finite volume quotients of H6
with |χ (M6)| < n grows as ncn for sufficiently large n, where c is a constant > 0.
Furthermore, for both the smooth finite volume quotients ofH3 obtained by Thurston’s
construction, and for the arithmetic quotients considered in the following subsection,
the vast majority of the smooth finite volume quotients are in fact compact, so it seems
likely that the number of topologically distinct smooth compact quotients of H6 with
|χ (M6)| < n also grows as ncn for sufficiently large n, with the same constant c > 0.
However, since supergravity in eleven dimensions does not contain any Yang-Mills
fields, and the three-form gauge field only enters the supercovariant derivative on the
gravitino through its three-form field strength, which is well defined globally, there
is no possibility of introducing an analogue of a spinc structure to compensate for
the compact six-manifold M6 not being a spin manifold, so M6 is required to be a
spin manifold. To the best of my knowledge, none of the known examples of smooth
compact quotients of CH3 or H6 have yet been shown to be spin manifolds.
For smooth compact quotients of CH3, the fraction of smooth compact quotients
that are spin would be expected to be ∼ 2−B2 , where B2 is the second Betti number
of the quotient, since the second Steifel-Whitney class is the mod 2 reduction of the
first Chern class, hence the vanishing of the second Steifel-Whitney class requires that
h11 ∼ B2 integers be even. Now by a theorem of Gromov [46], there is a number β such
that for all smooth compact quotients M6 of CH3 or H6 all the Betti numbers of M6
are bounded by β |χ (M6)|. Thus if we suppose that the fraction of the smooth compact
quotients of H6 that are spin is also ∼ 2−B2 , then the result of Burger, Gelander,
Lubotzky, and Mozes stated above implies that for sufficiently large |χ (M6)|, there
will be smooth compact quotients M6 of H6 that are spin.
It is known that all smooth compact orientable manifolds of dimension ≤ 3 are
spin [318], and the Davis manifold [40], which is the simplest known smooth compact
quotient of H4, has been shown to be a spin manifold [41].
We note that most of the smooth compact hyperbolic threefolds associated by
Thurston’s construction with a given smooth non-compact finite volume quotient of
H3 will be highly inhomogeneous. In fact, by a theorem of Cheeger [319], summarized
recently in [320], if a sequence of Riemannian manifolds is such that there is a fixed
upper bound on the magnitudes of all sectional curvatures of all manifolds in the se-
quence, a fixed lower bound > 0 on the volumes of all manifolds in the sequence, and a
fixed upper bound on the diameters of all manifolds in the sequence, then the sequence
contains only a finite number of diffeomorphism types. Thus since the members of a
Thurston infinite sequence of topologically distinct smooth compact quotients of H3,
whose volumes tend to the volume of a finite volume cusped hyperbolic threefold, satisfy
the first two requirements of Cheeger’s theorem, they must violate the third require-
ment, which means that the diameters of the members of the sequence must increase
without limit. This suggests that the members of such a Thurston sequence develop
longer and longer spikes, that approximate more and more closely to the infinite cusps
of the finite volume cusped hyperbolic threefold, and that, moreover, the differences
in topology between the members of the sequence might become localized further and
further out along these spikes, where the spikes become narrower and narrower.
3.1 Smooth compact arithmetic quotients of CHn and Hn
I shall now outline the construction of some smooth compact arithmetic quotients of
CHn and Hn. The first step is to construct some cocompact arithmetic lattices in
SU(n,1) and SO(n,1), whose existence follows from section 12 of [34]. I shall then
briefly review Selberg’s lemma [321] for the case of these arithmetic lattices, which
states that certain finite index subgroups of these discrete groups are torsion-free, or
in other words, have no nontrivial finite subgroups. A subgroup Γ1 of a discrete group
Γ is said to have finite index in Γ, if Γ1 divides Γ into a finite number of left cosets.
We recall that an algebraic number field [322] is a finite-dimensional (and therefore
algebraic) field extension of the field Q of rational numbers. That is, it is a field which
contains Q and has finite dimension, or degree, when considered as a vector space
over Q. To form an algebraic number field, we recall that for any field F , the ring of
polynomials with coefficients in F is denoted by F [x]. A polynomial p(x) in F [x] is
called irreducible over F [323] if it is non-constant and cannot be represented as the
product of two or more non-constant polynomials from F [x]. Then if α is a root of
some irreducible polynomial f (x) in F [x], the extension field F (α) is the set of all
polynomials g (α), with two polynomials g (α) and h (α) being defined to be equal, if
f (α) = 0 implies g (α) = h (α). In practice this means that, if f (x) is of degree m,
every polynomial g (α), of degree ≥ m, is equal to some polynomial of degree < m.
The field extension F (α) will then be of degree m, as a vector space over the field F ,
and a possible basis for F (α) is the set of monomials 1, α, α2, . . . , αm−1. If an element
a 6= 0 of F (α) is expressed in this basis as a = a1 + a2α+ a3α2 + . . .+ amαm−1, where
the coefficients ai are elements of F , then the reciprocal of a, expressed in this basis as
a−1 = b1 + b2α+ b3α
2 + . . .+ bmα
m−1, where the coefficients bi are elements of F , can
be found by solving the 2m− 1 linear equations, that result from equating coefficients
of all powers of α, up to and including α2m−2, in the equation
a1 + a2α + a3α
2 + . . .+ amα
b1 + b2α+ b3α
2 + . . .+ bmα
= 1 +
c1 + c2α + c3α
2 + . . .+ cm−1α
f (α) (412)
for the m coefficients bi, and the m − 1 coefficients ci, which are also elements of F .
The extension field F (α) is sometimes written F ({α}), to allow for the possibility of
adjoining more than one new element to F . In general, if S is a set of elements not in
F , the extension field F (S) is the smallest field that contains F and S.
We recall, also, that an algebraic number is a root of a polynomial with integer
coefficients. For any algebraic number, α, there is a unique polynomial f (x) in Q [x],
such that f (x) is irreducible over Q, the coefficient of the highest power of x in f (x)
is equal to 1, and α is a root of f (x). This is called the minimal polynomial of α, and
the degree of this polynomial is called the degree of α. Every polynomial g (x) in Q [x],
such that g (α) = 0, is a multiple of the minimal polynomial of α. The roots of the
minimal polynomial of α, including α itself, are called the conjugates of α, and are all
distinct.
By the primitive element theorem [324], every algebraic number field F is of the
formQ (α), where α is a root of a polynomial f (x) inQ [x], such that f (x) is irreducible
over Q. An element α of F , such that F is generated by adjoining α to F , is called a
primitive element of F . A primitive element of F can also be characterized by the fact
that it does not belong to any proper subfield of F , and it can also be characterized
by the fact that the degree of its minimal polynomial is equal to the degree of F . An
algebraic number field F has only a finite number of subfields K such that Q ⊆ K ⊆ F ,
and since these correspond to subspaces of F as a vector space over Q, most elements
of F are in fact primitive elements.
Considering the field F = Q
, for example, neither
2 nor
3 is a
primitive element of F , since they are respectively elements of the subfields Q
. We cannot form F by adjoining to Q a root of the polynomial x4−5x2+6,
whose roots are ±
2 and ±
3, because this polynomial factors as (x2 − 2) (x2 − 3),
and is thus not irreducible overQ. The nfinit function of PARI/GP [274], for example,
simply rejects an attempt to form an algebraic number field with this polynomial.
However α =
3, whose minimal polynomial is x4 − 10x2 + 1, is a primitive
element of F , and, indeed, we have α
2, and −α3
+ 11α
3. The conjugates
of α are ±
3 with all four sign choices allowed.
If F is an algebraic number field of degree m over Q, and vi, 1 ≤ i ≤ m, is
a basis for F , as a vector space over Q, then we may associate to each element x
of F , a square matrix xij with rational elements, defined by xvi = vjxji, where the
summation convention is used. We then find, for any elements x and y of F , that
xyvi = xvjyji = vjxjkyki. Thus the matrices xij form a matrix representation of the
elements of F , called the regular representation for the basis given by the vi. For
example, for the field Q
, we find that 1 is represented by
, and
is represented by
. Invariants of the matrix xij representing an element x
of F , such as its trace, determinant, and characteristic polynomial, are properties of
x, and do not depend on the basis. In particular, the characteristic polynomial of
xij is a polynomial of degree m, with the coefficient of λ
m equal to 1, and by the
Cayley-Hamilton theorem has x as a root. If x is a primitive element of F , then the
characteristic polynomial if its matrix representation, xij , is irreducible over Q, and is
the minimal polynomial of x.
We recall, also, that an algebraic integer [325] is a root of a polynomial with integer
coefficients, such that the coefficient of the highest power of x is equal to 1. The sum,
difference and product of two algebraic integers is an algebraic integer. If F is an
algebraic number field of degree m over Q, then the elements x of F , whose regular
representation matrices xij have a characteristic polynomial with integer coefficients,
are the algebraic integers in F . The set of all the algebraic integers in F is a ring, called
the ring of algebraic integers of F , that is often denoted OF . It is always possible to use
a basis for F consisting of algebraic integers, called an integral basis, in which every
algebraic integer x is represented by a matrix xij with integer matrix elements. When
we use an integral basis for F , the algebraic integers of F are then precisely those
elements x of F which, when expressed as a linear combination x = x1v1 + . . .+ xmvm
of elements of the basis, are such that the xi are all integers. Most of the algebraic
integers of F are primitive elements of F , since those which are not primitive are one
of the finite number of proper subfields K such that Q ⊆ K ⊂ F , and thus lie in one
of a finite number of linear spaces over Q, each of dimension < m over Q. Thus F can
always be expressed in the form Q (α), where α is an algebraic integer in F , of degree
equal to m.
If F is an algebraic number field of degree m over Q, and α is an algebraic integer
of F , of degree equal to m, then F always has an integral basis whose first element is 1,
whose second element has the form a21+α
, whose third element has the form a31+a32α+α
and so on, where the aij and the di are ordinary integers in Z. This is called a canonical
basis of F [326].
If t is an integer not divisible by the square of an integer > 1, then for t = 1 (mod4),
an integral basis for the quadratic number field Q
is given by 1, 1
, while
for t = 2 or 3 (mod 4), an integral basis for Q
is given by 1,
t [327].
If F is an algebraic number field, then an embedding of F into the field C of complex
numbers, sometimes called an isomorphism of F into C, means a one to one map of
F into C, that preserves all the structure of F . In particular, the subfield Q of F is
mapped by the identity map to the subfieldQ ofC. The number of distinct embeddings
of F into C is finite. In particular, if F is defined by adjoining to Q a root, α, of a
polynomial f (x) in Q [x], such that f (x) is irreducible over Q, then an embedding of
F = Q (α) into C is specified by saying which root of f (x), in C, α corresponds to.
An algebraic number field F is called totally real , if every embedding of F into C,
is equivalent to its complex conjugate. If F is defined by adjoining to Q a root, α, of
a polynomial f (x) in Q [x], such that f (x) is irreducible over Q, then F = Q (α) is
totally real, if all the roots of f (x) are real.
If an object X , such as a number, matrix, or group, is defined for a specific embed-
ding, I, of an algebraic number field, F , into C, then the object corresponding to X ,
for an embedding σ of F into C, is denoted by Xσ, and called the Galois conjugate
of X by σ. For example, if F is Q
, then there is only one embedding of F into
C different from I, and taking the Galois conjugate of an object by that embedding,
corresponds to replacing all occurrences of
2 by −
The discreteness of the arithmetic lattices, to be constructed below, will follow from
the fact, noted in section 12 of [34], that if F is an algebraic number field, OF is the
ring of algebraic integers of F , and ΦF is the set of all distinct embeddings σ of F
into C, then for any positive number r, there are only a finite number of elements b of
OF , such that all the Galois conjugates bσ of b, σ ∈ ΦF , have magnitude less than r.
For example, in the case when F = Q
and OF = Z
, there are an infinite
number of elements a + b
2, a, b ∈ Z, of OF , whose magnitude is less than 1, but
no elements at all of OF , such that both a + b
2 and a − b
2 have magnitude less
than 1. To check the result for general F , we note, first, that for any member, b, of
F , the set of all the Galois conjugates bσ of b, σ ∈ ΦF , is the same as the set of all
the conjugates of b, as an algebraic number. This set will have the same number of
members as the degree of F , if b is a primitive element of F , and a smaller number, if
b is not a primitive element of F . Furthermore, the product of all the conjugates of an
algebraic number of degree s is equal to (−1)s times the constant term in its minimal
polynomial, which must be non-zero, for otherwise the polynomial would be reducible
over Q. Thus the product of all the conjugates of an algebraic integer has magnitude
≥ 1, hence the result is certainly true when r ≤ 1, since there are then no elements of
OF , all of whose Galois conjugates have magnitude less than r. For general r, we note
that if all the conjugates of an algebraic integer of degree s have magnitude less than
r, then, denoting this algebraic integer and its conjugates by b1, b2, . . . , bs, we have
|b1 + b2 + . . .+ bs| < sr, |b1b2 + b1b3 + . . .+ bs−1bs| < s(s−1)2 r
2, . . ., |b1b2 . . . bs| < rs,
hence every coefficient of the minimal polynomial of that algebraic integer is bounded
by a binomial coefficient times a power of r, hence since these coefficients are integers,
there are, in fact, only a finite number of distinct algebraic integers of degree s, all
of whose conjugates have magnitude less than r. Hence, since all elements of OF are
algebraic integers of degree ≤ the degree of F , there are only a finite number of elements
of OF , all of whose Galois conjugates, or equivalently, all of whose conjugates, have
magnitude less than r. I will call this result the bounded conjugates lemma.
We can now construct examples of cocompact lattices in SU(n,1) and SO(n,1), by
choosing:
1. A totally real algebraic number field F 6= Q;
2. A specific embedding I of F into R; and
3. A diagonal (n+ 1) × (n+ 1) matrix B, with signature (+,+, . . . ,+,−), and
diagonal matrix elements in the ring of algebraic integers OF of F , such that for
all embeddings σ 6= I of F into R, the Galois conjugate Bσ is either positive
definite or negative definite.
For example, we could choose F to be Q (α), where α is a root of the polynomial
x2 − 2, we could specify I by choosing α to be
2 rather than −
2, and we could
choose B to be the diagonal (n + 1) × (n + 1) matrix with diagonal matrix elements
1, 1, . . . , 1,−
, so that for the one embedding σ different from I, the Galois conju-
gate Bσ is the positive definite diagonal matrix with diagonal matrix elements
1, 1, . . . , 1,
We note that SU (B), the group of all complex (n + 1)× (n+ 1) matrices with unit
determinant, that preserve the quadratic form BRS̄z
RzS̄, is isomorphic to SU(n,1), and
that the matrix ηRS̄, defined between (73) and (75), could be transformed to equal
BRS̄ , by a suitable rescaling of the coordinates, and similarly, SO (B), the group of
all real (n + 1)× (n+ 1) matrices with unit determinant, that preserve the quadratic
form BRSx
RxS, is isomorphic to SO(n,1), and that the standard Minkowski metric ηRS
could be transformed to equal BRS, by a suitable rescaling of the coordinates. Here
zS̄ =
, in accordance with the conventions of subsection 2.2.
If we now identify G with the identity component, or in other words, the connected
component containing the identity, of either SU (B) or SO (B), then the required co-
compact lattice, Γ, is in the unitary case, the subgroup GOF [i] of G, consisting of the
elements of G, all of whose matrix elements are in OF [i], the extension of the ring of
algebraic integers OF of F , by the square root of −1, and in the orthogonal case, the
subgroup GOF of G, consisting of the elements of G, all of whose matrix elements are
in OF [34].
To check the discreteness of Γ I will consider the case of SU (n, 1) and SU(B),
since the corresponding discussion for SO(n,1) and SO(B) will follow by dropping the
extension of OF to OF [i]. We first note that, for all embeddings σ 6= I of F into R, the
Galois conjugate group Gσ = SU (Bσ) is isomorphic to SU (n+ 1), and thus compact.
Furthermore, for any element γ of Γ, the Galois conjugate γσ will be a member of Γσ,
and thus of Gσ. Thus γσ could be transformed to a unitary matrix by a rescaling of
the coordinates, so there is a number rσ, depending only B
σ, such that every matrix
element of γσ has magnitude less than rσ. Let r be any number ≥ the maximum of
the numbers rσ, for all the embeddings σ 6= I of F into R. Then since OF is the ring
of algebraic integers of the algebraic number field F , the bounded conjugates lemma
implies that there are only a finite number of elements b of OF , such that all the
Galois conjugates bσ of b, including b itself, have magnitude less than r. Thus if γ is an
element of Γ, such that every matrix element (a+ ib), a, b ∈ OF , of γ, has magnitude
less than r, then there are only a finite number of elements c of OF [i], that can be
matrix elements of γ. Thus there are only a finite number of elements γ of Γ, such
that every matrix element of γ has magnitude less than r. In particular, there is a
number r > 0 such that there are only a finite number of elements γ of Γ, such that
every matrix element of γ − 1, where 1 denotes the unit matrix, has magnitude less
than r. Hence Γ is discrete [34].
We note that if r is the maximum of the numbers rσ, for all the embeddings σ 6= I
of F into R, and γ is any element of Γ, then every matrix element (a+ ib) of γ has the
property that each of a and b is an algebraic integer in OF , such that all its conjugates,
other than itself, have magnitude less than r. An algebraic integer, such that all its
conjugates, other than itself, have magnitude less than r, is called an r-Pisot number.
Pisot numbers are sometimes called Pisot-Vijayaraghavan numbers, or PV numbers.
Fan and Schmeling [328] have demonstrated that for any real algebraic number field,
and any r > 0, there exists a number L such that for all x ∈ R, there is at least
one r-Pisot number η in that algebraic number field, such that x ≤ η ≤ x + L. This
result gives an indication of the distribution of algebraic integers that can be matrix
elements of an element of Γ. For example, if F is the field Q
, G is isomorphic
to SU(1,1), and B is the diagonal matrix with diagonal entries
, then every
matrix element of every element of Γ is a 2
4 -Pisot number. Now for every integer, b,
there is an integer, a, such that
∣a− b
∣ ≤ 1
4 , so that a + b
2 is a 2
4 -Pisot
number. And for that integer, a, we have 2b
2 − 1
≤ a + b
2 ≤ 2b
2 + 1
. Thus in
this case we can take L = 2
2 + 1. Some examples of elements of Γ, in this case, are:
3 + 2
2 4 + 2
2 + 2
2 3 + 2
33 + 24
2 40 + 28
28 + 20
2 33 + 24
 (413)
Let PF,r denote the set of all the r-Pisot numbers in OF . We note that, for all real
numbers s > 0, there are only a finite number of elements of PF,r with magnitude < s.
For it is sufficient to prove this for s ≥ r. And for s ≥ r, every element of PF,r is
an s-Pisot number in F . And by definition, all the conjugates σ 6= I, of an s-Pisot
number, have magnitude < s. Hence by the bounded conjugates lemma, there are only
a finite number of s-Pisot numbers in F , whose magnitude is < s, hence there are only
a finite number of elements of PF,r, whose magnitude is < s. Hence, in particular, PF,r
is discrete.
3.1.1 Compactness of G/Γ for the examples of arithmetic lattices
The compactness of G/Γ, for groups such as those in the examples given above, was
originally proved by Borel and Harish-Chandra [34], making use of their proof that Γ is
a lattice in G, and their proof of a compactness criterion that had been conjectured by
Godement. The following direct proof of compactness is adapted from sections (6.36)
and (6.45) of [296], and the proof of Mahler’s compactness theorem [329] in section
(5.34) of [296].
To check the compactness of the quotient G/Γ, it is sufficient to check that, given
any infinite sequence {gk} of elements of G, there exists a sequence {γk} of elements
of Γ, such that the sequence {gkγk} has a convergent infinite subsequence.
We first note that if G = SU (B) ∼= SU (n, 1), the subgroup GF (i) of G, consisting of
the elements of G, all of whose matrix elements are in F (i), the extension of F , by the
square root of −1, is dense in G, while if G = SO (B) ∼= SO(n, 1), the subgroup GF of
G, consisting of the elements of G, all of whose matrix elements are in F , is dense in
G. For an arbitrary element V of SU(B) satisfies
V †BV = B (414)
where V † denotes the hermitian conjugate of V , and (414) is also satisfied by an
arbitrary element V of SO (B), since in that case V † = V T , where V T denotes the
transpose of V . And in general, if B is a nonsingular hermitian matrix, and V is a
complex matrix that satisfies (414), then the matrix
A ≡ B (V − 1) (V + 1)−1 (415)
is antihermitian, and V is expressed rationally in terms of the antihermitian matrix A
V = (B − A)−1 (B + A) (416)
Furthermore, the matrix (B − A) = 2B (V + 1)−1 is nonsingular. Moreover, if A is an
arbitrary antihermitian matrix, such that (B − A) is nonsingular, and V is defined in
terms of A by (416), then V satisfies V †BV = B. Thus, for an arbitrary element V of
SU (B) or SO (B), such that (V + 1) is non-singular, we can define the antihermitian
matrix A by (415), and then, by choosing an antihermitian matrix Ã, with matrix
elements in F (i) or F , as appropriate, that approximates A sufficiently closely, and
is such that
B − Ã
is nonsingular, we can obtain an element Ṽ of SU (B)F (i) or
SO (B)F , as appropriate, such that every matrix element of
Ṽ − V
has magnitude
less than any given number > 0. And if (V + 1) is singular, we can follow the same
procedure, for an element V1 of SU (B) or SO (B), as appropriate, such that that matrix
elements of (V1 − V ) are sufficiently small in magnitude, and (V1 + 1) is nonsingular, so
as to obtain, again, an element Ṽ of SU (B)F (i) or SO (B)F , as appropriate, such that
every matrix element of
Ṽ − V
has magnitude less than any given number > 0. Thus
it is sufficient to check that, given any infinite sequence {gk} of elements of GF (i) or GF ,
as appropriate, there exists a sequence {γk} of elements of Γ, such that the sequence
{gkγk} has a infinite Cauchy subsequence, or in other words, an infinite subsequence
{gpγp}, such that for any given ε > 0, there exists an integer t, such that for all p > t
and all q > t, every matrix element of (gpγp − gqγq) has magnitude less than ε.
The matrices γk will be constructed as one block of a block diagonal matrix that
includes all the Galois conjugates of Γ along its block diagonal, because we can then
transform these block diagonal matrices to matrices with integer matrix elements, by a
similarity transformation that consists of multiple copies of the inverse of the similarity
transformation that diagonalizes the matrix representations of the elements of F in an
integral basis, discussed above. Once we are working with matrices with integer matrix
elements, we can construct the sequence {γk} by a method due to Mahler [329].
It is convenient, first, if G = SU (B) ∼= SU (n, 1), to embed G in a group of
2 (n+ 1) × 2 (n + 1) matrices with real matrix elements. For each element g of G,
let ḡ denote the 2 (n+ 1)× 2 (n + 1) matrix with real matrix elements, obtained from
g by replacing each complex matrix element (a+ ib) by the real matrix
We note that, by this rule, the hermitian conjugate g† of g corresponds to the transpose
ḡT of ḡ. And let B̄ be obtained from B by the same rule. Thus B̄ is a diagonal matrix
whose first and second diagonal matrix elements are equal to one another, whose third
and fourth diagonal matrix elements are equal to one another, and so on. B̄ has sig-
nature (+,+, . . . ,+,−,−), so we are embedding G in a group isomorphic to SO (n, 2).
However the following discussion will not depend on the detailed signature of B̄ or B,
beyond the fact that B̄ or B is indefinite, while its Galois conjugates for σ 6= I are
either positive definite or negative definite.
When an element g of SU (B) acts on a complex (n + 1)-vector, each complex matrix
element (c+ id) of that (n+ 1)-vector is replaced by the real column vector
, so
ad− bc
bc+ ad
, and the complex (n + 1)-vector becomes a real
2 (n+ 1)-vector. These two ways of representing a complex number, as a real matrix,
or as a real column vector, are an example of the relation between the representation
of an element x, of an algebraic number field, by the matrix xij , and by its components
xi, in a particular basis, as discussed above.
We note that if ḡ corresponds to an element g of SU (B) by the transformation
described above, then each 2 × 2 block in ḡ can be diagonalized by the similarity
transformation 1√
a+ ib 0
0 a− ib
, hence
all the 2 × 2 block matrices in ḡ can be diagonalized by applying a block diagonal
similarity transformation with n + 1 copies of
 along the block diagonal.
Then by permuting rows and columns, ḡ can be brought to the form of a block diagonal
matrix with two (n+ 1) × (n+ 1) blocks along the block diagonal, one of which is g,
and the other of which is g∗, the complex conjugate of g. Hence det ḡ = |det g|2. But
det g = 1, hence det ḡ = 1.
Let J be the block diagonal matix with n copies of
 along the block
diagonal. Then the subgroup SU (B) of SO
is the group of all elements g of SO
such that J g = gJ .
Hence by renaming B̄ as B, we can now assume that either B is a diagonal matrix
with n positive diagonal matrix elements and one negative diagonal matrix element, and
G = SO(B), or B is a diagonal matrix with 2n positive diagonal matrix elements and
2 negative diagonal matrix elements, and G is the subgroup of SO (B) that commutes
with J . And for every Galois conjugate σ of F different from the identity I, Bσ is in
both cases either a positive definite matrix or a negative definite matrix. We define
p = (n + 1) in the orthogonal case, and p = 2 (n + 1) in the unitary case.
Let H be the set of all p × p matrices with matrix elements in F , and m be the
degree of F . We now choose a fixed sequence of the m Galois conjugates σ of F ,
starting with the identity σ = I, and for an arbitrary element h of H , we define ĥ to be
the pm× pm block diagonal matrix which has h as its top left p× p matrix elements,
then the first Galois conjugate σ 6= I of h as its second block of p× p matrix elements
on the block diagonal, then the second Galois conjugate σ 6= I of h as its third block of
p× p matrix elements on the block diagonal, and so on, and all other matrix elements
equal to zero. We also define Ĥ to be the set of all the matrices ĥ, for h ∈ H , Ĝ to
be the set of all the matrices ĝ, for g ∈ GF , and Γ̂ to be the set of all the matrices ĝ,
for g ∈ Γ. We note that if g is any element of GF , then since ĝ is block diagonal, and
every block on the block diagonal of ĝ has determinant equal to 1, det ĝ = 1.
For an arbitrary p-vector x in F p, we define x̂ to be the pm-vector whose first p
components are x, whose next p components are the first Galois conjugate σ 6= I of x,
whose third set of p consecutive components are the second Galois conjugate σ 6= I of
x, and so on. We also define F̂ p to be the set of all the pm-vectors x̂, for x ∈ F p. Thus
for all ĥ ∈ Ĥ , and all x̂ ∈ F̂ p, the pm-vector ĥx̂ is an element of F̂ p. We also define
ÔpF to be the set of all the pm-vectors x̂, for x ∈ O
We note that, if x is any nonzero p-vector in F p, then the quadratic form xTBx
is nonzero. For by assumption F 6= Q, hence F has at least one Galois conjugate σ
different from I, and by assumption (xσ)
Bσxσ =
is either positive definite or
negative definite. Furthermore, no nonzero element of F can have any Galois conjugate
equal to 0, for 0 is of degree 1, hence has no conjugates other than itself. Thus if x̂ is
any nonzero element of F̂ p, then the quadratic form x̂T B̂x̂ is nonzero.
Furthermore, if x is any nonzero p-vector in OpF , then the value of the quadratic
form x̂T B̂x̂ is an ordinary integer in Z, and its magnitude is ≥ 1. For all the matrix
elements of B are in OF , hence xTBx is an algebraic integer in OF . Furthermore,
x̂T B̂x̂ is the sum of all the Galois conjugates
of xTBx, which if the degree
of the algebraic integer xTBx is equal to m, is −1 times the coefficient of xm−1 in the
minimal polynomial of xTBx, and thus an integer in Z, while if the degree k of xTBx
is less than m, it must divide m, and x̂T B̂x̂ is equal to the integer m
, times −1 times
the coefficient of xk−1 in the minimal polynomial of xTBx, and thus again an integer
in Z. And furthermore, by the preceding paragraph, the ordinary integer x̂T B̂x̂ cannot
be equal to zero, hence it has magnitude ≥ 1.
We now carry out a similarity transformation ĥ → S−1ĥS on the elements ĥ of
Ĥ , such that S consists of p copies of the inverse, s, of a similarity transformation
that diagonalizes the matrix representations of the elements of F in an integral basis,
discussed above. Each of the p copies of s is “spread out”, so that, for example, the first
copy acts from the right only on the first column of each of the m Galois conjugates
of elements of GF , the second copy acts from the right only on the second column
of each of the m Galois conjugates of elements of GF , and so on. For example, if
F = Q
, we can choose the similarity transformation S−1ĥS to consist of p copies
of the similarity transformation:
0 a− b
 (417)
And if, in addition, G ∼= SO (1, 1), so that p = 2, the similarity transformation S−1ĥS
would have the form:
1 0 1 0
0 − 1√
0 1 0 1
0 − 1√
2 c+ d
2 0 0
e + f
2 g + h
2 0 0
0 0 a− b
2 c− d
0 0 e− f
2 g − h
2 0 0
0 0 1
2 0 0
0 0 1 −
a 2b c 2d
b a d c
e 2f g 2h
f e h g
(418)
We see that for each element h of H , the similarity transformation S−1ĥS transforms
ĥ into a p × p block matrix, each block of which is the m ×m matrix representation
of the corresponding matrix element of h, in the chosen integral basis. Thus the
matrix elements of S−1ĥS are rational numbers. For each element h of H , we define
h̃ ≡ S−1ĥS, where ĥ is the element of Ĥ that corresponds to h as above. We see that
the elements of the subgroup Γ̂ of Ĝ are precisely those elements ĝ of Ĝ for which
g̃ = S−1ĝS has integer-valued matrix elements. We define H̃ to be the set of all the
matrices h̃, for h ∈ H , G̃ to be the set of all the matrices g̃, for g ∈ GF , and Γ̃ to be
the set of all the matrices g̃, for g ∈ Γ. We note that if g is any element of GF , then
since g̃ is related to ĝ by a similarity transformation, and det ĝ = 1, we have det g̃ = 1.
Now if α is a primitive element of F , or in other words, an algebraic number in
F , whose degree is equal to m, and Qm×m denotes the set of all m×m matrices with
rational matrix elements, then the elements of Qm×m that are matrix representations
of elements of F , in the chosen integral basis, are precisely those that commute with
the matrix representation of α in the chosen integral basis. For every element of F
commutes with α, and if an element χ of Qm×m commutes with the matrix repre-
sentation of α, then since (1, α, α2, . . . , αm−1) is a possible basis for F , so the matrix
representations of 1, α, α2, . . . , αm−1 are linearly independent elements of Qm×m, and
are thus a complete basis for the elements of Qm×m that commute with α, χ is a linear
combination, with coefficients in Q, of the matrix representations of 1, α, α2, . . . , αm−1,
and is thus the matrix representation of an element of F .
We now choose an algebraic integer ϕ of F , such that ϕ is primitive in F . Such an
algebraic integer ϕ of F always exists, because, as noted above, most algebraic integers
in F are primitive in F . We define F to be the element of H that is the p× p diagonal
matrix, with each matrix element on the diagonal equal to ϕ, so that, in other words,
F is equal to ϕ times the p×p unit matrix. The elements F̂ of Ĥ, and F̃ of H̃ , are then
defined in the standard way, as above. Thus F̃ is the pm× pm block diagonal matrix,
such that each m×m block on the block diagonal is equal to the matrix representation
of ϕ, in the chosen integral basis. Then if Qpm×pm denotes the set of all pm × pm
matrices with matrix elements in Q, the elements of H̃ are precisely the elements of
Qpm×pm that commute with F̃ , since an element ξ of Qpm×pm commutes with F̃ if and
only if every m ×m block of ξ commutes with the matrix representation of ϕ, in the
chosen integral basis.
We note that B̃ will have integer-valued matrix elements, and be block diagonal,
with each block on the block diagonal being an m × m matrix which, for at least
one block, in the example when F is Q
, will not be symmetric. F̃ will also have
integer-valued matrix elements, since ϕ is an algebraic integer. And in the unitary case,
J̃ will have integer-valued matrix elements, which will in fact be +1 or −1, and be
block diagonal, with each block on the block diagonal being a 2m× 2m antisymmetric
matrix.
We note that in the above example, (417), s has been chosen such that every matrix
element in its first column is equal to 1. This has the consequence that for an arbitrary
element h of H , the first column of the matrix ĥS is the element x̂ of F̂ p, where x
denotes the first column of h, and the element x̂ of F̂ p is related to the p-vector x in
F p, as described above. Furthermore, the first m matrix elements of the first column
of S−1ĥS, or in other words, the first m matrix elements of S−1x̂, are the components
of the first matrix element of x, with respect to the integral basis
and the next m matrix elements of the first column of S−1ĥS, or in other words, the
next m matrix elements of S−1x̂, are the components of the second matrix element
of x, with respect to the integral basis
. Therefore, since for an
arbitrary p-vector x in F p, we can write down an element h of H , such that the first
column of h is x, it follows, for this example, that for an arbitrary p-vector x in F p, the
components of the pm-vector S−1x̂, where the element x̂ of F̂ p is related to the p-vector
x in F p as described above, are the m components of the first matrix element of x,
with respect to the integral basis
, followed by the m components of
the second matrix element of x, with respect to the integral basis
Thus, for this example, for an arbitrary p-vector x in OpF , the pm-vector S−1x̂ has
integer components in Z, and conversely, for an arbitrary pm-vector x̃ in Zpm, the
pm-vector Sx̃ is an element x̂ of ÔpF , that corresponds to an element x of O
F in the
manner described above. Specifically, we have:
1 0 1 0
0 − 1√
0 1 0 1
0 − 1√
a + b
(419)
This corresponds to the fact that, with s chosen as in the example (417), s−1 acts on a
column m-vector, that consists of all the Galois conjugates of an element of F , in the
standard order, as:
a + b
 (420)
I shall now show that for an arbitrary algebraic number field F , and thus, in particular,
for an arbitrary totally real algebraic number field F , we can always choose an integral
basis {vi} for F , such that the elements of the first column of the matrix representation
of each element of F , in the basis {vi}, are the expansion coefficients of that element of
F in the basis {vi}, 1 ≤ i ≤ m. And furthermore, in such a basis for F , the inverse, s, of
the similarity transformation that diagonalizes the matrix representations of elements
of F in the integral basis {vi}, will be such that all the matrix elements in its first
column are nonzero, and moreover, all the matrix elements in its first column can be
chosen equal to 1.
When we use such a basis for F , and choose all the matrix elements in the first
column of s to be equal to 1, then it immediately follows, as in the above example,
that for an arbitrary p-vector x in OpF , the pm-vector S−1x̂ has integer components in
Z, which are in fact the expansion coefficients of the successive matrix elements of x,
with respect to the integral basis {vi}, and conversely, for an arbitrary pm-vector x̃ in
Zpm, the pm-vector Sx̃ is an element x̂ of ÔpF , that corresponds to an element x of O
in the manner described above.
We choose an integral basis vi, 1 ≤ i ≤ m, for F , such that v1 = 1. For example, we
could choose a canonical integral basis, associated with the algebraic integer ϕ, in terms
of which we defined the matrices F , F̂ , and F̃ above. We recall, from the beginning
of this subsection, that the matrix elements xij of the representation of an element x
of F , for the basis given by the vi, are defined by xvi = vjxji, where the summation
convention is used. Thus for the basis element vk we find vkv1 = vk = vj (vk)j1, hence
by the linear independence of the basis elements, we must have (vk)j1 = δkj . On the
other hand, we can also express a general element x of F , in the integral basis vi, as
x = xkvk. Hence xi1 = xk (vk)i1 = xi. Thus the elements of the first column of xij are
the expansion coefficients of x in the integral basis vi.
Let s be the inverse of the similarity transformation that diagonalizes the matrix
representations of elements of F in the integral basis vi, so that S consists of p copies
of s, each “spread out”, as described above. Thus if x is a general element of F , and x̂
is the diagonal m×m matrix, whose matrix elements on the diagonal are the m Galois
conjugates of x, taken in the same order as we chose above, then (s−1x̂s)ij = xij , where
xij are the matrix elements of x in the basis vi, which by assumption has v1 = 1. Now
this equation remains true, with the same x̂ and xij , if we pre-multiply s by an arbitrary
diagonal matrix, so that s−1 gets post-multiplied by the inverse of that diagonal matrix.
Thus by pre-multiplying s by a suitable diagonal matrix, we can assume that every
nonzero matrix element, in the first column of s, is equal to 1. Furthermore, no matrix
element in the first column of s can be zero. For by assumption, each matrix element
of the first column of x̂s is either the appropriate Galois conjugate of x, or zero. And
by the preceding paragraph, the set of the first columns of (s−1v̂is), for the m basis
elements vi, is a set of m linearly independent column vectors of real numbers, namely
, . . . ,
. But this would be impossible, if any matrix element of
the first column of s was zero, because the set of the first columns of (v̂is) would not
then be a set of m linearly independent column vectors of real numbers. Thus we can
assume that every matrix element, in the first column of s, is equal to 1, as in the
example above.
We now choose all the matrix elements in the first column of s to be equal to 1,
so that for an arbitrary p-vector x in OpF , the pm-vector S−1x̂ has integer components
in Z, which are the expansion coefficients of the successive matrix elements of x, with
respect to the integral basis {vi}, and for an arbitrary pm-vector x̃ in Zpm, the pm-
vector Sx̃ is an element x̂ of ÔpF , that corresponds to an element x of O
F in the manner
described above.
We next note that by choosing each matrix element in the first column of s to be
equal to 1, we have guaranteed that the matrix V ≡ sT s has rational matrix elements.
For by the definition of s, we have, for an arbitrary element x of F , that x̃s−1 = s−1λ,
where λ is a diagonal matrix. Hence x̃ = s−1λs, and x̃T = sTλ (s−1)
= V x̃V −1, hence
(V x̃)
= V x̃, or in other words, V x̃ is a symmetric matrix. If we now regard V as
an independent symmetric matrix, and impose this condition on V , for an arbitrary
primitive element x of F , then since all the eigenvalues of x̃ are distinct for x primitive,
we find 1
m (m− 1) linearly independent equations among the matrix elements of V ,
with coefficients linear in the matrix elements of x̃, and thus rational numbers. For if
we regard V as an independent symmetric matrix, and express the equation x̃TV =
V x̃ in a basis in which x̃ is diagonal, or in other words, if we write the equation
as λ (s−1)
V s−1 = (s−1)
V s−1λ, and treat this as an equation for the symmetric
matrix V , without making use of the relation between s and V , then the fact that λ
is a diagonal matrix, all of whose eigenvalues are different, implies that the equation
x̃TV = V x̃ is equivalent to 1
m (m− 1) linearly independent relations among the matrix
elements of the symmetric matrix V , of the form (λi − λj)
(s−1)
V s−1
= 0, 1 ≤
i < j ≤ m, where the summation convention is not applied to i and j. But the number
of linearly independent relations in the matrix equation x̃TV = V x̃ is independent of
what basis we express it in, hence for x primitive this matrix equation gives 1
m (m− 1)
linearly independent linear relations, with rational coefficients, among the 1
m (m+ 1)
independent matrix elements of the symmetric matrix V , which we can use to express
m (m− 1) matrix elements of V as linear combinations, with rational coefficients, of
the remaining m independent matrix elements. Furthermore, the form of the equation,
in the basis in which all the x̃ are diagonal, which simply states that the symmetric
matrix (s−1)
V s−1 is diagonal, shows that no further information can be obtained, by
imposing the relation x̃TV = V x̃, for any further x̃.
We choose to use the relation x̃TV = V x̃, for one primitive x, to express all the
m (m− 1) independent Vij , 2 ≤ i ≤ m, 2 ≤ j ≤ m, as linear combinations, with
rational coefficients, of the Vi1, 1 ≤ i ≤ m. And making use, now, of the definition
V = sT s, and the fact that we have set si1 = 1, for all 1 ≤ i ≤ m, we find that:
m s12 + s22 + . . .+ sm2 . . . s1m + s2m + . . .+ smm
s12 + s22 + . . .+ sm2 s
12 + s
22 + . . .+ s
32 . . . s12s1m + . . .+ sm2smm
. . .
s1m + s2m + . . .+ smm s12s1m + . . .+ sm2smm . . . s
1m + s
2m + . . .+ s
(421)
Thus it remains to check that the linear combinations s12 + s22 + . . .+ sm2, s13 + s23 +
. . .+ sm3, ..., s1m + s2m + . . .+ smm, are rational numbers. To do this, we use the fact
that sT is the matrix of eigenvectors of the x̃, or in other words, for all x in F , we have
x̃T sT = sTλ, where λ is the diagonal matrix of eigenvalues of x̃T . We have chosen the
top matrix element of each column of sT to be equal to 1. The sums whose rationality
we want to determine, are the sums of the matrix elements across the rows 2 to m of
sT . In other words, for each row of sT after the first, we need to check that the sum of
all the matrix elements in that row of sT is rational. We choose a primitive element x
of F , so that the eigenvalues λi of x̃
T are all distinct, and for each column i of sT , use
rows 2 to m of the equation x̃T sT = sTλ, to express each matrix element after the first
of that column of sT , or in other words, the sij, 2 ≤ j ≤ m, as a ratio of multinomials
formed from the matrix elements of x̃T , the eigenvalue λi for that column of s
T , and
the top matrix element of that column of sT , which is 1. When we do this for all the
columns i of sT , 1 ≤ i ≤ m, we find that for all the sij in each row j of sT , we have a
formula of the form
sij =
fj (λi)
gj (λi)
(422)
where fj (λi) and gj (λi) are multinomials in the matrix elements of x̃
T and λi, such
that the dependence of fj (λi) and gj (λi) on the matrix elements of x̃
T is the same for
all the sij in the row j of s
T . For example, for m = 3 we find, for row 2 of sT , that:
s12 = −
x̃12 (λ1 − x̃33) + x̃13x̃32
λ1x̃33 + x̃22 (λ1 − x̃33) + x̃23x̃32 − λ21
s22 = −
x̃12 (λ2 − x̃33) + x̃13x̃32
λ2x̃33 + x̃22 (λ2 − x̃33) + x̃23x̃32 − λ22
s32 = −
x̃12 (λ3 − x̃33) + x̃13x̃32
λ3x̃33 + x̃22 (λ3 − x̃33) + x̃23x̃32 − λ23
(423)
From the structure of the formula (422), we see that
i=1 sij is a symmetric function
of the eigenvalues λi, and can in fact be expressed as the ratio of two symmetric
multinomials in the λi. Thus it is equal to the ratio of two multinomials in the matrix
elements of x̃, so it is a rational number. Thus the matrix elements of the matrix
V = sT s are rational numbers.
Now the quadratic form xTBx is preserved by all elements g of GF , for by definition
of G, we have gTBg = B, for all elements g of GF . And similarly, we have ĝ
T B̂ĝ = B̂,
for all elements ĝ of Ĝ, because this equation is block diagonal, with each of the m
blocks on the block diagonal being one of the m Galois conjugates of the first p × p
block on the block diagonal, which is an equation of the form gTBg = B, with g ∈ GF .
We now define the symmetric matrix B̆ = ST B̂S = STSB̃, where B̃ = S−1B̂S as
above, and the S is constructed from p “spread out” copies of s, as described above.
Then since the matrix elements of B̃ are ordinary integers, and the matrix elements of
STS are rational numbers, the matrix elements of B̆ are rational numbers. We next
note that, for all elements g̃ of G̃, we have:
g̃T B̆g̃ =
S−1ĝS
ST B̂S
S−1ĝS = ST ĝT B̂ĝS = ST B̂S = B̆ (424)
Now let x̃ be an arbitrary nonzero pm-vector with integer components, or in other
words, an arbitrary nonzero element of Zpm. Then x̂ = Sx̃ is a nonzero element of ÔpF ,
that corresponds to a nonzero element x of OpF in the manner described above. Hence,
from above, the value of x̃T B̆x̃ = (S−1x̂)
ST B̂S
S−1x̂ = x̂T B̂x̂ is an ordinary integer
in Z, and its magnitude is ≥ 1. The fact that the set of possible values of x̃T B̆x̃ is
discrete, and that there is a minimum distance > 0 between adjacent possible values
of x̃T B̆x̃, also follows directly from the fact that the matrix elements of B̆ are rational
numbers.
We next note that there is a real number λ > 0, such that for all g ∈ GF , and all
nonzero pm-vectors x̃ ∈ Zpm, |g̃x̃| ≥ λ. Here, and throughout the following, |g̃x̃| has its
usual meaning of |g̃x̃| =
x̃T g̃T g̃x̃. For if |g̃x̃| could be arbitrarily small, then the value
of x̃T g̃T B̆g̃x̃ = x̃T B̆x̃ could be arbitrarily close to 0. But by the preceding paragraph,
the magnitude of x̃T B̆x̃ is ≥ 1, for arbitrary nonzero x̃ ∈ Zpm. Hence for nonzero
x̃ ∈ Zpm, x̃T B̆x̃ cannot be arbitrarily close to 0, hence |g̃x̃| cannot be arbitrarily small.
Let λ > 0 be the largest number such that for all g ∈ GF , and all nonzero pm-vectors
x̃ ∈ Zpm, |g̃x̃| ≥ λ.
Given an infinite sequence {gk} of elements of GF , the plan now is to find, first, a
sequence {βk} of elements of SL (pm,Z), such that the sequence {g̃kβk} has an infinite
Cauchy subsequence {g̃jβj}, and then show that this infinite Cauchy subsequence itself
has an infinite subsequence {g̃iβi}, such that βiβ−11 is an element γ̃i of Γ̃, for all i in
this infinite subsequence. The sequence {ĝiγ̂i} = {Sg̃iγ̃iS−1} is then an infinite Cauchy
sequence of block diagonal matrices in Ĝ, such that all the γ̂i are in Γ̂, and the sequence
{giγi} of the first p× p blocks, on the block diagonal, is an infinite Cauchy sequence in
GF , such that the sequence {gi} is an infinite subsequence of the given sequence {gk},
and all the γi are in Γ.
To construct the required sequence βk of elements of SL (pm,Z), we first use a
method of Mahler [329] to construct, for an arbitrary element g of GF , an element β
of SL (pm,Z), such that all matrix elements of g̃β are bounded above in magnitude in
terms of λ, where λ > 0 was defined above to be the largest number such that for all
g ∈ GF , and all nonzero pm-vectors x̃ ∈ Zpm, |g̃x̃| ≥ λ. The following form of Mahler’s
construction is adapted from section (5.34) of [296]. We define l ≡ pm.
Given an element g of GF , the required element β of SL (l,Z) will be constructed
column by column, as a sequence of nonzero column vectors in Zl, that I shall call
v1, v2, v3, . . . , vl.
We first choose v1 ∈ Zl\{0}, where \ means “outside”, such that |g̃v1| has its mini-
mum possible value, for v ∈ Zl\{0}. This is always possible, because g̃ is nonsingular,
hence vT g̃T g̃v is a positive definite quadratic form, with no flat directions. Let π1
denote the projection to the line Rg̃v1, and π
1 denote the projection to the subspace
orthogonal to this line.
We next choose v2 ∈ Zl\Rv1, such that
∣π⊥1 g̃v2
∣ has its minimum possible value, for
v ∈ Zl\Rv1. This is always possible, for the same reason as before. Moreover, π⊥1 g̃v2 is
unaltered by adding a multiple of v1 to v2. For k ∈ Z, the values of |π1g̃ (v2 + kv1)| =
g̃v1(vT1 g̃
T g̃(v2+kv1))
vT1 g̃
T g̃v1
= |g̃v1|
vT1 g̃
T g̃v2
vT1 g̃
T g̃v1
are spaced by |g̃v1|, so by replacing v2 by
v2 + kv1, with a suitable value of k, we can assume that |π1g̃v2| ≤ 12 |g̃v1|. Then from
the minimality of |g̃v1|, we have that:
|g̃v1| ≤ |g̃v2| ≤
∣π⊥1 g̃v2
∣+ |π1g̃v2| ≤
∣π⊥1 g̃v2
|g̃v1| (425)
Hence:
∣π⊥1 g̃v2
|g̃v1| (426)
Let π2 denote the projection to the plane Rg̃v1 +Rg̃v2, and π
2 denote the projection
to the subspace orthogonal to this plane.
We next choose v3 ∈ Zl\(Rv1 +Rv2), such that
∣π⊥2 g̃v3
∣ has its minimum possible
value, for v ∈ Zl\(Rv1 +Rv2). This is always possible, for the same reason as before.
Moreover, π⊥2 g̃v3 is unaltered by adding multiples of v1 and v2 to v3. We first add a
suitable integer multiple of v2, to arrange that
1 g̃v3
∣ ≤ 1
∣π⊥1 g̃v2
∣. Then, without
affecting this bound, we add a suitable integer multiple of v1, to arrange that |π1g̃v3| ≤
|g̃v1|. The minimality of
∣π⊥1 g̃v2
∣ now implies:
∣π⊥1 g̃v2
∣π⊥1 g̃v3
∣π⊥2 g̃v3
1 g̃v3
∣π⊥2 g̃v3
∣π⊥1 g̃v2
∣ (427)
Hence:
∣π⊥2 g̃v3
∣ ≥ 1
∣π⊥1 g̃v2
∣ (428)
Then we carry on after this pattern, until we eventually choose
vl ∈ Zl\(Rv1 + . . .+Rvl−1), such that
∣π⊥l−1g̃vl
∣ has its minimum possible value, for
v ∈ Zl\(Rv1 + . . .+Rvl−1). Then by successively adding suitable integer multiples of
vl−1, vl−2, ..., v2, and v1, we arrange that
∣πl−1π
l−2g̃vl
∣ ≤ 1
∣π⊥l−2g̃vl−1
∣πl−2π
l−3g̃vl
∣π⊥l−3g̃vl−2
∣, ..., |π1g̃vl| ≤ 12 |g̃v1|. Then from the minimality of
∣π⊥l−2g̃vl−1
∣, we find, in
the same way as before, that:
∣π⊥l−1g̃vl
∣ ≥ 1
∣π⊥l−2g̃vl−1
∣ (429)
We next note that the successive minimality of |g̃v1|,
∣π⊥1 g̃v2
∣π⊥2 g̃v3
∣, ...,
∣π⊥l−1g̃vl
implies in turn that the convex hull of {0, v1} contains no points of Zl other than 0
and v1, the convex hull of {0, v1, v2} contains no points of Zl other than 0, v1, and v2,
..., and finally that the convex hull of {0, v1, v2, . . . , vl} contains no points of Zl other
than 0, v1, v2, ..., vl. Hence the parallelepiped generated by the vectors v1, v2, ..., vl,
whose vertices are the expressions of the form k1v1 + k2v2 + . . . + klvl, where each ki
can independently take the values 0 or 1, contains no points of Zl in its convex hull,
other than its 2l vertices.
Now by considering tesselations of Rl by lattice parallelepipeds, the volume of a
lattice parallelepiped is given, in terms of the points of Zl in its convex hull, by:
f2 + . . .+
fl−1 + fl, (430)
where f0 is the number of points of Z
l that are vertices of the parallelpiped, f1 is the
number of points of Zl that lie within the “interiors” of edges of the parallepiped, ...,
and fl is the number of points of Z
l that lie within the interior of the l-volume of the
parallepiped. Hence in the present instance, the volume of the parallelepiped generated
by the vectors v1, v2, ..., vl, is 1, hence the determinant of β, which is defined to be the
l× l matrix whose columns are v1, v2, ..., vl, is ±1. And if det β = −1, we note that we
can replace vl by −vl, which is also in Zl\(Rv1 + . . .+Rvl−1), without affecting the
minimality of
∣π⊥l−1g̃vl
∣, and we therefore replace vl by −vl, to obtain det β = 1.
Furthermore:
∣π⊥l−1g̃vl
∣π⊥l−2g̃vl−1
∣π⊥l−3g̃vl−2
∣ . . .
∣π⊥1 g̃v2
∣ |g̃v1| = |det (g̃β)| = 1 (431)
Hence from (426), (428), ..., and (429), we find that:
|g̃v1| ≤ 2
2 (432)
Furthermore, since v1 6= 0, we have |g̃v1| ≥ λ > 0. Therefore, returning to (431), and
the bounds (426), (428), ..., we find that
∣π⊥1 g̃v2
∣ ≤ 2
∣π⊥2 g̃v3
∣ ≤ 2
. . .
∣π⊥k−1g̃vk
∣ ≤ 2
(k−2)(k−1)
2(l+1−k) 2
l+1−k
. . .
∣π⊥l−1g̃vl
∣ ≤ 2
(l−2)(l−1)
(433)
Furthermore, since |π1g̃v2| ≤ 12 |g̃v1|, we find:
|g̃v2| ≤
∣π⊥1 g̃v2
∣+ |π1g̃v2| ≤ 2
2 (434)
And similarly:
|g̃v3| ≤
∣π⊥2 g̃v3
1 g̃v3
∣+ |π1g̃v3| ≤
2 (435)
And so on. Thus, since λ > 0, all matrix elements of g̃β are, indeed, bounded, inde-
pendently of g̃.
Given an infinite sequence {gk} of elements of GF , we now take, for each k, βk
to be the matrix β, as constructed above, with g̃ taken as g̃k. The elements of the
sequence {g̃kβk} are then bounded in terms of λ as above, independently of k. Hence
this sequence has a Cauchy subsequence. We can find a Cauchy subsequence by subdi-
viding the bounded l2-dimensional domain of the matrix elements into a finite number
of subsectors, choosing a subsector in which the sequence has an infinite number of
elements, subdividing that subsector into an finite number of subsectors, choosing one
of them in which the sequence has an infinite number of elements, and so on. Let
{g̃jβj} be an infinite Cauchy subsequence of the sequence {g̃kβk}.
Now we found above that the matrix elements of B̆ are rational numbers. Hence
there is an integer a ∈ Z such that all the matrix elements of aB̆ are ordinary integers
in Z. On the other hand, the fact that {g̃jβj} is a Cauchy sequence implies that the
sequence
βTj g̃
j B̆g̃jβj
βTj B̆βj
(436)
is a Cauchy sequence. Hence since all the matrix elements of aβTj B̆βj are ordinary
integers in Z, there must be a value q1 of j such that for all s ≥ q1 and all t ≥ q1,
aβTs B̆βs = aβ
t B̆βt, hence
B̆βsβ
t = B̆. This result can also be obtained
without directly using the fact that the matrix elements of B̆ are rational numbers, by
using the fact that for an arbitrary element x̃ of Zl, the value of x̃T B̆x̃ is an ordinary
integer in Z, and considering the Cauchy sequence
x̃TβTj g̃
j B̆g̃jβj x̃
x̃TβTj B̆βj x̃
for 1
l (l + 1) suitable choices of x̃, such as the l unit vectors in the positive coordinate
directions, and the 1
l (l − 1) distinct sums of two such unit vectors. This procedure
can also be used to give an alternative proof that the matrix elements of B̆ are rational
numbers, without using the fact that the matrix elements of sT s are rational numbers.
Furthermore, the fact that {g̃jβj} is a Cauchy sequence implies that the sequence
β−1j g̃
j F̃ g̃jβj
β−1j F̃βj
(437)
is a Cauchy sequence. Hence since the matrix elements of F̃ are ordinary integers in
Z, hence the matrix elements of β−1j F̃βj are ordinary integers in Z, there must be
a value q2 of j such that for all s ≥ q2 and all t ≥ q2, β−1s F̃βs = β−1t F̃βt, hence
F̃βsβ−1t = βsβ−1t F̃ .
And finally, in the unitary case, the preceding paragraph is also valid with F̃ re-
placed by J̃ , hence there must be a value q3 of j such that for all s ≥ q3 and all t ≥ q3,
J̃ βsβ−1t = βsβ−1t J̃ .
Hence there is a value r of j, namely the maximum of q1 and q2 in the orthogonal
case, and the maximum of q1, q2, and q3 in the unitary case, such that for all s ≥ r and
all t ≥ r, βsβ−1t is an element of G̃. Let γ̃j ≡ βjβ−1r for all j ≥ r, and let {g̃iβi} be the
infinite Cauchy sequence obtained from {g̃jβj} by dropping all terms with j < r. Then
{g̃iβiβ−1r } = {g̃iγ̃i} is an infinite Cauchy sequence in G̃, such that {g̃i} is an infinite
subsequence of {g̃k}, and γ̃i ∈ G̃, for all i. Then as anticipated above, the sequence
{ĝiγ̂i} = {Sg̃iγ̃iS−1} is an infinite Cauchy sequence of block diagonal matrices in Ĝ,
such that all the γ̂i are in Γ̂, and the sequence {giγi} of the first p × p blocks, on the
block diagonal, is the required infinite Cauchy sequence in GF , such that the sequence
{gi} is an infinite subsequence of the given sequence {gk}, and all the γi are in Γ.
3.1.2 Obtaining finite index torsion-free subgroups of Γ by Selberg’s lemma
We recall from above that for compact quotients of SU (n, 1) or SO (n, 1), the re-
quirement that the quotient be smooth, rather than an orbifold, or in other words,
that all elements 6= 1 of the discrete subgroup Γ act on the symmetric space CHn =
SU (n, 1) /(SU (n)× U (1)) or Hn = SO(n, 1) /SO (n) without fixed points, is equiva-
lent to the requirement that Γ have no torsion, or in other words, no nontrivial finite
subgroups. Selberg’s lemma [321], for the case of arithmetic lattices such as those in
the examples above, states that certain finite index subgroups of these discrete groups
have no torsion. We recall that a subgroup Γ1 of a discrete group Γ is said to have finite
index in Γ, if Γ1 divides Γ into a finite number of left cosets. Thus if G/Γ is compact,
and Γ1 has finite index in Γ, then G/Γ1 is also compact, so for all the examples above,
we can obtain smooth compact quotients of CHn or Hn by using any of the subgroups
of Γ specified by Selberg’s lemma for this case. I shall briefly review Selberg’s lemma
for the case of these arithmetic lattices, following section (5.60) of [296].
As in the previous subsection, we define p = 2 (n + 1) if G = SU (n, 1), and p = n+1
if G = SO(n, 1), and l ≡ pm, where m is the degree of F . We choose an integral basis
for F , and represent each element γ of Γ as a p× p block matrix, each block of which
is the m × m matrix representation of the corresponding matrix element of γ, in the
chosen integral basis. Thus each element γ of Γ is represented by an l× l matrix, with
matrix elements in Z. In the preceding subsection, such l × l matrices representing
elements γ of Γ were denoted γ̃, and the group of all of them was denoted Γ̃, but
since Γ̃ is isomorphic to Γ, and this representation of Γ, as a group of l × l matrices,
with matrix elements in Z, is the only representation of Γ that will be used in the
present subsection, I shall not use the tildes in this subsection. Thus we now regard
Γ as a subgroup of SL (l,Z). The following construction is valid for all subgroups Γ of
SL (l,Z).
For k ∈ Z, such that k ≥ 2, let Γk denote the set of all elements of Γ of the form
(1 + kT ), where 1 denotes the l× l unit matrix, and the matrix elements of T are in Z.
Then Γk is a group, and is moreover a normal subgroup of Γ, since if γ is an element
of Γ, then γ is a matrix with matrix elements in Z, and determinant equal to 1, so
γ−1 (1 + kT ) γ has the form 1 + kT1, where the matrix elements of T1 are in Z.
We next note that elements γ1 and γ2 of Γ are in the same left left coset of Γk in
Γ, if and only if corresponding matrix elements of γ1 and γ2 are equal, mod k. For if
corresponding matrix elements of γ1 and γ2 are equal, mod k, then γ2 = γ1 + kT , for
some matrix T with matrix elements in Z, hence γ−11 γ2 = 1 + kγ
1 T , which is in Γk,
so γ1 and γ2 are in the same left coset of Γk in Γ, while if γ1 and γ2 are in the same
left coset of Γk in Γ, then γ2 = γ1 (1 + kT ) for some matrix T with matrix elements in
Z, hence corresponding matrix elements of γ1 and γ2 are equal, mod k.
Thus the quotient group Γ/Γk, of Γ by its normal subgroup Γk, is the group obtained
from Γ, by considering its matrix elements mod k. Thus each matrix element of Γ/Γk
takes values in the finite set {0, 1, 2, . . . , k − 1}, hence Γ/Γk cannot have more than
2) elements, and is thus a finite group, and Γk has finite index in Γ.
We now demonstrate that for k ≥ 3, Γk has no torsion. It is sufficient to demonstrate
that for an arbitrary element γ of Γk, such that γ 6= 1, no integer power s ≥ 1 of γ is
equal to 1, for if γ is an element of a finite group, the sequence {1, γ, γ2, γ3, . . .} cannot
contain more distinct elements than the number of elements of that finite group.
We assume now that k ≥ 3. Then k is divisible by either 22 or an odd prime.
Furthermore, Γj is a subgroup of Γk whenever k is a divisor of j, so it is sufficient to
prove that Γk has no torsion when k is either 2
2 or an odd prime. Thus we now assume
k = pr, where p is prime, and r = 2 for p = 2, and r = 1 for p ≥ 3. Furthermore, it is
sufficient to prove that for an arbitrary element γ of Γk, such that γ 6= 1, no power γs
is equal to 1 for s prime, since if s factorizes as s = tq, where q is prime, we can write
γs = (γt)
. Thus we now assume s is prime, so either p does not divide s, or p = s.
We can write a general element of Γk as (1 + p
uT ), where u ≥ r ≥ 1, and not
every matrix element of T is divisible by p. If p does not divide s, we note that (puT )
is equal to 0, mod pu+1, hence (1 + puT )
is equal to 1 + spuT , mod pu+1, which is
6= 1, mod pu+1. And if p = s, we note that (puT )3 is equal to 0, mod pu+2, hence
(1 + puT )
= 1+pu+1T +
p2u+1(p−1)
T 2, mod pu+2. Furthermore, if p ≥ 3, then (p− 1) is
even, hence
p2u+1(p−1)
is an integer that is equal to 0, mod pu+2, hence (1 + puT )
p 6= 1,
mod pu+2, while if p = 2, then u ≥ 2, hence again p
2u+1(p−1)
is an integer that is equal
to 0, mod pu+2, hence (1 + puT )
p 6= 1, mod pu+2.
An alternative method of constructing a torsion-free subgroup of Γ has been consid-
ered by Everitt and Maclachlan [42], who applied their method to obtain a construction
of the Davis manifold [40], which is the smallest known smooth compact quotient of
3.2 Smooth compact quotients of S3
For even n, every real antisymmetric (n+ 1) × (n + 1) matrix has a zero eigenvalue,
hence no element of SO (n + 1) acts without fixed points on Sn, and the only smooth
compact quotient of Sn is the non-orientable n-dimensional real projective space, ob-
tained from Sn by identifying every point with its antipode. On the other hand, S3
is well known to have smooth compact quotients, which fall into a small number of
families, that were first classified by Seifert and Threlfall [330, 331]. Smooth com-
pact quotients of S3 have been considered recently as possible topologies for the three
observed spatial dimensions, in consequence of the current slight preference of astro-
physical data for k = +1 rather than k = −1, as discussed in section 2, and have
recently been reclassified by Gausmann, Lehoucq, Luminet, Uzan, and Weeks [332].
4 The Casimir energy densities
The validity of the realization of TeV-scale gravity by the thick pipe geometries studied
in section 2, for the compactification of Hořava-Witten theory on a particular smooth
compact quotient of CH3 or H6 that is a spin manifold, and a particular choice of spin
structure on that spin manifold, depends on the Casimir energy densities on and near
the inner surface of the thick pipe resulting in the integration constant B, in (206),
taking the value (313), or (399), and in the case where the outer surface of the thick
pipe is stabilized by Casimir effects, also on the Casimir energy densities on and near
the outer surface resulting in the integration constant Ã, in (265), taking the value
(315). The Casimir energy densities are, by definition, the correction terms in the field
equations and boundary conditions for the graviton, when they are derived by varying
the full quantum effective action, or in other words, the generating functional of the
proper vertices, with respect to the graviton field, rather than by varying the classical
Cremmer-Julia-Scherk action, augmented by supersymmetrized Gibbons-Hawking [93,
94, 95, 69, 70, 71] terms and the semiclassical Hořava-Witten supersymmetric Yang-
Mills actions on the orbifold fixed-point hyperplanes.
Now, as noted in connection with (129), on page 57, the quantum effective action
Γ (Φ), for a properly gauge-fixed classical action A (ϕ), where ϕ denotes all the fields
occurring in the gauge invariant classical action, together with all the Faddeev-Popov
ghosts [228, 227, 230, 231], and also the Nielsen-Kallosh ghosts [234, 235] if appropriate,
can be calculated, for an arbitrary classical field configuration Φ, as the sum of all
the one line irreducible vacuum diagrams, calculated from the action A (Φ + ϕ), with
the term linear in ϕ deleted, where ϕ denotes the quantum fields. In other words,
using DeWitt’s compact index notation [227], where a single index, i, runs over all
combinations of type of field, space-time position, and coordinate and other indices,
the quantum effective action, as a function of the classical fields, Φ, is given by the sum
of all the one line irreducible vacuum diagrams, calculated with the action A (Φ + ϕ)−
δA(Φ)
, as in (129), where the summation convention is applied to the index i.
To check this, we note that, with the functional integral defined as Z (J) = e−iW (J) =
[dϕ] exp i (A (ϕ) + Jiϕi), the classical field Φi defined as Φi ≡ − δWδJi , and Γ (Φ) defined
by a Legendre transformation by the relation Γ (Φ) + JiΦi = −W (J), we have Ji =
−ιii δΓ
, where ιij is −1 if both ϕi and ϕj are fermionic, and 1 otherwise, indices on
ιij are ignored in applying the summation convention, and all derivatives act from the
left [125, 126, 333]. We then have [334]:
eiΓ(Φ) =
[dϕ] ei(A(ϕ)+Ji(ϕi−Φi)) =
[dϕ] e
A(ϕ)−(ϕi−Φi) δΓδΦi
(438)
which can be regarded as an alternative definition of Γ (Φ). Shifting the integration
variables ϕi by Φi, we have:
eiΓ(Φ) =
[dϕ] e
A(Φ+ϕ)−ϕi δΓδΦi
[dϕ] e
A(Φ+ϕ)−ϕi δAδΦi
(439)
where the loop expansion Γ (Φ) = A (Φ)+Γ1 (Φ)+Γ2 (Φ)+ . . . was introduced. Now if
the term −i
+ . . .
in the exponent in the right-hand side of (439) was
neglected, (439) would express Γ (Φ) as the sum of all connected, but not necessarily
one line irreducible, vacuum bubbles, calculated with the action
A (Φ + ϕ)− ϕi δA(Φ)δΦi
An arbitrary such vacuum bubble can be regarded as a tree diagram, such that a
vertex of the tree diagram on which n propagators end corresponds to i
δnΓ(Φ)
δΦi1 ...δΦin
, and
a propagator of the tree diagram corresponds to iG (Φ)i1i2 , where G (Φ)ij is the inverse
of the matrix
δ2A(Φ)
δΦiδΦj
And when the effects of the term −i
+ . . .
in the exponent are
included, the only change is that each vertex, of a tree diagram, on which precisely
one propagator ends, can now come from either of two alternative sources, namely
either as a one line irreducible diagram built from the propagators G (Φ)ij and vertices
δnA(Φ)
δΦi1 ...δΦin
, n ≥ 3, of the action
A (Φ + ϕ)− ϕi δA(Φ)δΦi
, as before, or alternatively from
the term of appropriate loop order ≥ 1 in the extra term −i
+ . . .
. The
result of this is that a tree diagram that contains m vertices i δΓ
, on which precisely
one propagator ends, gets a factor (1− 1)m. Hence since every tree diagram with
more than one vertex contains at least one such vertex, all the tree diagrams cancel
out except for those with precisely one vertex, and these are the one line irreducible
vacuum bubbles calculated with the action
A (Φ + ϕ)− ϕi δA(Φ)δΦi
, as stated.
Continuing to use DeWitt’s abstract index notation, the one-loop effective action,
Γ1 (Φ), is given, by (439), by:
eiΓ1(Φ) = K1
[dϕ] e
δ2A(Φ)
δΦjδΦi
superdet
δ2A(Φ)
δΦjδΦi
= K2e
supertr ln
δ2A(Φ)
δΦjδΦi (440)
where K1 and K2 are constants independent of the fields Φi, and the matrix
δ2A(Φ)
δΦjδΦi
has been assumed to have a bose-bose part
δ2A(Φ)
δΦjδΦi
, for which the indices j and i
denote bosonic fields, and a fermi-fermi part
δ2A(Φ)
δΦjδΦi
, for which the indices j and i
denote fermionic fields, but no non-vanishing matrix elements such that one of the
indices i and j is bosonic, and the other fermionic, in which case the superdeterminant
[335, 336, 337] is defined by:
superdet
δ2A (Φ)
δΦjδΦi
δ2A(Φ)
δΦjδΦi
δ2A(Φ)
δΦjδΦi
(441)
and the bose-bose part
δ2A(Φ)
δΦjδΦi
has been assumed to have an infinitesimal positive-
definite imaginary part.
We assume now that A (Φ) has an expansion:
A (Φ) =
AijΦjΦi +
AijkΦkΦjΦi +
AijklΦlΦkΦjΦi + . . . (442)
and define G (Φ)ij to be the inverse of the matrix
δ2A(Φ)
δΦjδΦi
, and Gij to be the inverse of the
matrix Aij. Then since A (Φ) is bosonic, and the assumed vanishing of all bose-fermi
matrix elements of
δ2A(Φ)
δΦjδΦi
thus implies that all non-vanishing matrix elements of
δ2A(Φ)
δΦjδΦi
are bosonic, we have:
G (Φ)ij = Gij −Gik
δ2A (Φ)
δΦkδΦm
− Akm
δ2A (Φ)
δΦkδΦm
− Akm
δ2A (Φ)
δΦnδΦp
− Anp
Gpj − . . . (443)
And from (440) we have, up to an additive constant, independent of all the fields Φi:
ιjjGjk
δ2A (Φ)
δΦkδΦj
ιjjGjk
δ2A (Φ)
δΦkδΦl
δ2A (Φ)
δΦmδΦj
+ . . .
(444)
where the effect of the factors ιjj is to introduce a − sign when the field circulating
in the loop is fermionic. The expression (444) is well-known to be real, in Minkowski
signature. Checking it is real is simplest for the field equations. We have:
δ3A (Φ)
δΦiδΦkδΦj
Gjk −Gjm
δ2A (Φ)
δΦmδΦn
− Amn
Gnk + . . .
δ3A (Φ)
δΦiδΦkδΦj
ιjjG (Φ)jk (445)
Thus, from the definition (14) of the energy-momentum tensor, the one-loop Casimir
energy density contributions to the energy-momentum tensor are obtained from this
equation, by choosing the field Φi to be the metric, gµν , and multiplying by
Considering, now, the contribution to the one-loop Casimir energy densities from a
real scalar boson, with the classical action:
Ascalar = −
gµν (Dµϕ) (Dνϕ) +m
2ϕ2 − ξRϕ2
(446)
where R is the Ricci scalar, and ξ is a real constant, sometimes called the conformal
coupling, when d = 4, we find, by use of the Palatini identity δRµν = DµδΓ
τν−DτδΓτµν ,
and the identity δΓτµν =
gτσ (Dµδgσν +Dνδgσµ −Dσδgµν), that:
δAscalar
δΦgxµν
(1− 2ξ) gµσgντ (Dσϕ) (Dτϕ)−
gµνgστ (Dσϕ) (Dτϕ)
−2ξgµσgντϕDσDτϕ+ 2ξgµνgστϕDσDτϕ− ξ
Rµν − 1
ϕ2 − 1
gµνm2ϕ2
(447)
For the particular case ξ = 1
, this is in agreement with the formulae of Muller, Fagun-
des, and Opher [236, 238], after for allowing for their sign convention for the Riemann
tensor, which results in the opposite sign of the Ricci tensor to (8).
From (14), (445), and (447), we find:
1,scalar =
δΓ1,scalar
δΦgxµν
(1− 2ξ) gµσgντ (DyσDxτ +DxσDyτ )
gµνgστ (DyσDxτ +DxσDyτ )− 2ξgµσgντ (DxσDxτ +DyσDyτ )
+2ξgµνgστ (DxσDxτ +DyσDyτ )− 2ξ
Rµν −
− gµνm2
G (Φ)ϕxϕy
(448)
In the particular case of 3+ 1 dimensional Minkowski space, the scalar contribution to
the one-loop energy density is given by (448) as:
T 001,scalar =
(2− 4ξ) ∂y0∂x0 + (1− 4ξ)
−∂y0∂x0 + ~∂y~∂x
− 2ξ (∂x0∂x0 + ∂y0∂y0)
−∂2x0 + ~∂2x − ∂2y0 + ~∂2y
G (Φ)ϕxϕy
(449)
Furthermore, for the scalar propagator, requiring that the Fresnel integral in (440) be
well-defined uniquely selects the Feynman iε prescription for the propagator. Thus the
scalar propagator is:
Gϕxϕy = −
ei(−p0(x0−y0)+~p.(~x−~y))
−p20 + ~p2 +m2 − iε
(450)
Substituting (450) into (449), and taking the limit y → x from either y0 > x0 or
y0 < x0, we find:
T 001,scalar =
~p2 +m2 (451)
which is real, as required, and is the standard divergent expression for the one-loop
vacuum energy density of a real scalar field. For models that have unbroken super-
symmetry in 3 + 1 spacetime dimensions, and do not involve gravitons, the one-loop
vacuum energy densities cancel between fermions and bosons, and, moreover, the vac-
uum energy density is exactly zero to all orders in the coupling constants [338], and the
one-loop vacuum energy densities have also been found to vanish in some models with
broken supergravity [339, 340, 245, 246], whilst for d = 11 supergravity [14], it appears
that a cosmological constant is not consistent with supersymmetry [200, 269, 124], so
that divergences corresponding to a cosmological constant term would be cancelled
unambiguously within the framework of BPHZ renormalization, to all orders in the
semiclassical expansion in the number of loops in the Feynman diagrams.
Considering the gravitino propagator, for the compactification of Hořava-Witten
theory on M6, as a sum over images:
ψxµiψyνj
Φ|CH3
ψxµiψγ(y)νj
, (452)
where M6 is the quotient of CH3 by the cocompact, torsionless, discrete subgroup
Γ of SU (3, 1), we see that if the sign of the gravitino field, at the image γ (y) of y
by an element γ of Γ, depends on the route taken from y to γ (y), then the sum
of G (Φ|CH3)ψxµiψγ(y)νj , for y close to x, over all elements γ of Γ different from the
identity, will not be well defined, even if it converges. But this sum is directly physically
significant, because it determines the finite part of the gravitino contribution to the
one-loop Casimir contribution to the energy-momentum tensor, by a formula analogous
to (448). Furthermore, the three-form gauge field, CIJK , only enters the gravitino field
equation through its four-form field strength GIJKL, which is globally well defined, so
a background configuration of the three-form gauge field cannot make any difference
to whether or not the sum over γ ∈ Γ in G (Φ|M6)ψxµiψyνj well defined. Thus it does,
indeed, seem that models of this type are not physically well defined, unless M6 is a
spin manifold. Of course, a rotation through 2π changes the sign of a spinor field, so it
is natural to wonder whether introducing twists or rotations in the local Lorentz part of
the vielbein between different coordinate patches, which will cancel out of the relations
between the metric on the different coordinate patches, can cancel the ambiguity, but
this is presumably taken into consideration in determining whether or not a manifold
is a spin manifold. A direct explanation of why CP2 is not a spin manifold has been
given by Hawking and Pope [39], and recently reviewed in Appendix B of [341].
4.1 The Salam-Strathee harmonic expansion method
For the explicit calculation of the Casimir energy densities for compactifications on
smooth compact quotients of CH3 or H6, by means of the sum over images method
of Muller, Fagundes, and Opher [236, 237, 238], for obtaining the propagators on the
quotients, or some extension of their method if the sums diverge at large distances
due to the masslessness of the fields, the propagators and heat kernels for the d = 11
supergravity multiplet are needed for flat R5, times CH3 or H6, and the propagators
and heat kernels for the d = 10 supersymmetric Yang-Mills multiplet are needed for flat
R4, timesCH3 orH6. The propagators and heat kernels forCH3 orH6 can be obtained
from the corresponding propagators and heat kernels for CP3 or S6, which can in turn
be obtained by a straightforward but lengthy application of the harmonic expansion
method of Salam and Strathdee [244, 342, 343], which is currently in progress. The
harmonic expansions can be summed by means of a generating function, and for the
heat kernel of a massive scalar, on CH3, we find the integral representation:
H (χ, s) =
4e−sm
d cosh (2χ)
+ (cosh (2χ) + 1)
d cosh (2χ)
(dy) e−
4s sinh (2y) (cosh (2y)− cosh (2χ))−
2 (453)
Here χ is proportional to the geodesic distance between the two position arguments of
the heat kernel. The same integral, but with different differential operators acting on
it, occurs in the heat kernel of a massive scalar, on real hyperbolic spaces of all even
dimensions ≥ 2, while for real hyperbolic spaces of odd dimension ≥ 3, the heat kernel
can be written in closed form, as found by Muller, Fagundes, and Opher, for d = 3.
The application of the Salam-Strathdee method to CP3, which is a spin manifold, was
begun by Strathdee [343], and developed by Sobczyk [344, 345].
A special effect in a related background was discovered by Gibbons and Nicolai [340],
who calculated the one-loop vacuum energy density of the Freund-Rubin AdS4 × S7
compactification of d = 11 supergravity [346], and found that it vanished “floor by
floor”, or in other words, separately for each N = 8 supersymmetric Kaluza-Klein level,
whereas to preserve the supersymmetry of the vacuum, it would have been sufficient
for the sum over all the Kaluza-Klein levels to vanish. The contribution of the lowest
Kaluza-Klein level, namely the N = 8 supergravity multiplet, had earlier been found
to vanish by Allen and Davis [339].
There is also a AdS4 × CP3 compactification of type IIA d = 10 supergravity,
discovered by Watamura [347], that was shown by Nilsson and Pope [348] to have
N = 6 supersymmetry, and also to be related to the Freund-Rubin compactification
of d = 11 supergravity, via the fact that S7 is a non-trivial U (1) fiber bundle over
CP3, called a Hopf fibration. What this means is that the Watamura AdS4 × CP3
compactification of type IIA d = 10 supergravity can be identified with a particular
AdS4×CP3×S1 compactification of d = 11 supergravity, such that the metric ansatz
(94) has been modified by the replacement
dy2 →
dy −AAdzA
, (454)
where AA is proportional to a potential for the Kähler form of the CP
3, and y is
now the coordinate around the S1. The Watamura N = 6 compactification is then
obtained in an appropriate limit, where the radius of the S1 tends to 0, while for another
special case, where the radius of the S1 is appropriately related to the diameter of the
CP3, the supersymmetry is presumably extended to N = 8, and the Freund-Rubin
compactification is obtained.
Nilsson and Pope showed that the complete spectrum of small fluctuations of the
Watamura N = 6 compactification can be directly obtained from the known spectrum
of small fluctuations of the Freund-Rubin solution [349, 350, 351, 352, 353]. I shall
now obtain the complete list of the modes by the Salam-Strathdee method, and check
it against the list given by Nilsson and Pope, and then repeat the Gibbons-Nicolai
calculation, for all but the lowest two Kaluza-Klein levels, for the Watamura N = 6
compactification.
The isometry group of CP3, with the standard Fubini-Study metric [72], is SU (4),
and the subgroup of the isometry group, that leaves a chosen point fixed, which I
shall call the tangent space isometry group, is SU (3) × U (1). The tangent space
group of CP3 is SO (6), because CP3 has six real dimensions, so by accident, the
tangent space group is locally isomorphic to the isometry group, although the tangent
space group and the isometry group are completely distinct, and the tangent space
isometry group, SU (3) × U (1), is found to be embedded in the tangent space group,
SO (6), and the isometry group, SU (4), in different ways. I shall put a tilde above
the irreducible representations of the tangent space group, SO (6), to distinguish them
from the irreducible representations of the isometry group, SU (4).
The first step of the Salam-Strathdee method is to decompose all the fields in-
volved, which are here the metric, the three-form gauge field, and the gravitino, of
d = 11 supergravity, into irreducible representations of the product of the tangent
space isometry groups SO (3, 1), of the four extended dimensions, and SU (3)× U (1),
of CP3, and possible components along the S1, that does not have a nontrivial con-
tinuous tangent space isometry group. The next step is then to determine, for each
irreducible represention of the tangent space isometry group SU (3)×U (1) of CP3 that
arises, the list of all the irreducible representations of SU (4), the isometry group of
CP3, that contain that irreducible representation of SU (3)×U (1), under the subgroup
decomposition SU (4) → SU (3)×U (1). This is then the list of all the harmonics that
occur, in the harmonic expansion, on CP3, of that particular irreducible representation
of the tangent space isometry group of CP3.
According to Salam and Strathdee’s general prescription, [244], for harmonic ex-
pansions on the quotient space, G/H , of a Lie group, G, and a Lie subgroup, H , of
G, the quotient space CP3 = SU (4) /(SU (3)× U (1)) is coordinatized by the “boosts”
generated by the six generators of SU (4), that are not generators of its subgroup
SU (3)×U (1). Now SU (3)×U (1) is contained in SU (4) such that the 4 of SU (4) has
the SU (3)× U (1) content:
4 = 3 1
+ 1− 3
(455)
where the relative value of the U (1) charges, which are shown as subscripts, is deter-
mined by the tracelessness of the SU (4) generators, and the overall normalization of
the U (1) charges is a convention, that I have chosen to agree with Strathdee, [343],
and Sobczyk, [344, 345].
From (455), we find that the SU (3) × U (1) content of the adjoint of SU (4) is
determined by:
15 + 1 = 4× 4̄ =
+ 1− 3
3̄− 1
+ 1 3
= 80 + 10 + 31 + 3̄−1 + 10 (456)
Thus the generators of SU (4), that are not generators of SU (3) × U (1), have the
SU (3)×U (1) content 31+3̄−1, so the tangent space isometry group, SU (3)×U (1), of
CP3, is embedded in the tangent space group, SO (6), of CP3, such that the tangent
space six-vector, the 6̃ of SO (6), has the SU (3)× U (1) content [343]:
6̃ = 31 + 3̄−1 (457)
The decomposition (457) now determines the decompositions of all the other irreducible
representations of SO (6). In particular, if we consider the 4̃ of SO (6) that contains
the 1 and the 3̄ of SU (3), and write its decomposition as 4̃ = 1a + 3̄b, where the U (1)
charges a and b are to be determined, we find that:
4̃× 4̃ = 12a + 3̄a+b + 3̄a+b + 6̄2b + 32b (458)
However, we know that the antisymmetric part of 4̃× 4̃ is the 6̃, so for consistency with
(457), we must have 2b = 1, and a+ b = −1, so that we find 4̃ = 1− 3
+ 3̄ 1
. The other
4̃ of SO (6) then decomposes as 1 3
+ 3− 1
, so, comparing with (455), we see that the
tangent space isometry group, SU (3)×U (1), of CP3, is, indeed, embedded differently
in the tangent space group, SO (6), and the isometry group, SU (4), as stated above
[343].
To determine which irreducible representations of SU (4) contain a given irreducible
representation of SU (3)× U (1), it will be convenient to use Young tableau notations
for the irreducible representations of SU (3) and SU (4). I shall denote the irreducible
representation of SU (3), that corresponds to a Young tableau with rows of lengths a,
b, and c, such that a ≥ b ≥ c ≥ 0, by [a, b, c], with a corresponding notation for SU (4).
Then [a+ n, b+ n, c + n], for all integer n such that c + n ≥ 0, all correspond to the
same irreducible representation of SU (3), whose Dynkin label is (a− b, b− c).
It will be very convenient also to allow Young tableaux with negative length rows,
which means that a, b, and c, in the Young tableau [a, b, c], are restricted only by
a ≥ b ≥ c, without the restriction to c ≥ 0. Negative length rows are represented
by rows of blocks extending out to the left of what would normally be the left-hand
side of the Young tableau. The corresponding irreducible representations of SU (3) are
constructed from appropriately symmetrized Kronecker products of the fundamental
and the antifundamental representations, one fundamental representation factor for
each block in a positive length row, and one antifundamental representation factor for
each block in a negative length row, with all traces that can be formed by contracting
the SU (3) invariant tensor δrs̄ with an antifundamental index and a fundamental index,
both from among the left-hand indices of the representation matrix, or both from among
the right-hand indices of the representation matrix, removed.
Analogous constructions also apply for all the other special unitary groups. For
example, the [p, 0, 0,−p] representation of SU (4), whose Dynkin label is (p, 0, p), with
the three components of the Dynkin label corresponding to the three vertices of the
SU (4) Dynkin diagram, written in sequence from end to end along the line, has the
representation matrices:
UI1I2...IpJ̄1J̄2...J̄p,K̄1K̄2...K̄pL1L2...Lp =
(p! )
SISJSKSL
(−1)r
(2p+ 2− r) !
((p− r) ! )2 r! (2p+ 2) !
×δL1K̄1δI1J̄1 . . . δLrK̄rδIrJ̄rUIr+1K̄r+1UJ̄r+1Lr+1 . . . UIpK̄pUJ̄pLp
(459)
where, in the last line of this expression,
δL1K̄1δI1J̄1 . . . δLrK̄rδIrJ̄rUIr+1K̄r+1UJ̄r+1Lr+1 . . . UIpK̄pUJ̄pLp
is interpreted as having no δs, if r < 1, and as no Us, if r + 1 > p. In other words,
the subscripts, on the subscripts, are to increase from 1 to p, going from left to right,
along this expression. UIJ̄ here denotes the fundamental representation of an element
of SU (4), and UĪJ ≡ (UIJ̄)
denotes the antifundamental representation of that same
element of SU (4), in accordance with the conventions of subsection 2.2, on page 25,
for barred and unbarred indices, and
denotes the sum over all permutations of the
“I” subscripts, and thus contains p! terms, since there are p such subscripts, and so on.
The formula (459) is used in the derivation of the scalar heat kernel on CP3, (453), by
the Salam-Strathdee method.
We then find, using indices µ, ν, σ, . . ., for the four extended dimensions, r, s, t, . . .,
and r̄, s̄, t̄, . . ., in the complex coordinate notation of subsection 2.2, for CP3, and y
for the S1, that the d = 11 gravition, hIJ , contains the d = 4 graviton, hµν , in the
Young tableau representation [0, 0, 0]0 of SU (3) × U (1), where the subscript denotes
the U (1) charge, and also a d = 4 vector, hµy, and a d = 4 scalar, hyy, in the [0, 0, 0]0
of SU (3) × U (1), a d = 4 vector, hµr, and a d = 4 scalar, hyr, in the [1, 0, 0]1 of
SU (3) × U (1), a d = 4 vector, hµr̄, and a d = 4 scalar, hyr̄, in the [0, 0,−1]−1 of
SU (3) × U (1), and d = 4 scalars, hrs̄, in the [1, 0,−1]0, hrs, in the [2, 0, 0]2, and hr̄s̄,
in the [0, 0,−2]−2, of SU (3)×U (1). The decomposition of the three-form gauge field,
CIJK , is worked out similarly, bearing in mind that for d = 4, a two-form gauge field is
equivalent to a scalar [354, 355, 356, 233], and a three-form gauge field has no degrees
of freedom.
To work out the decomposition of the gravitino, we first determine the decomposi-
tion of a d = 11 spinor. We decompose the 32-valued spinor index into the Cartesian
product of an 8-valued spinor index, for the CP3, and a four-valued spinor index, for
the four extended dimensions and the S1, considered together as a five-dimensional
space, and consider the decomposition of the 8-valued spinor index, which is the sum
of the two opposite chirality 4̃’s of SO (6). Thus, from above, the 8-valued spinor index
decomposes into the 1− 3
+3̄ 1
+3− 1
= [0, 0, 0]− 3
+[0, 0,−1]1
+[0, 0, 0]3
+[1, 0, 0]− 1
of SU (3) × U (1). The d = 11 gravitino, ψI , thus contains d = 4 gravitinos, ψµ, and
d = 4 spinors, ψy, in these four representations of SU (3)× U (1), together with d = 4
spinors, ψr, in the SU (3)×U (1) representations that result from forming the Cartesian
product of the 31 with these four representations, namely 3
+3̄ 1
[1, 0, 0]− 1
+ [1, 0,−1] 3
+ [0, 0, 0]3
+ [1, 0, 0]5
+ [2, 0, 0]1
+ [0, 0,−1] 1
, and d = 4 spinors,
ψr̄, in the complex conjugates of these six representations.
To determine which irreducible representations of SU (4) contain these represen-
tations of SU (3) × U (1), we first recall the general rule for the irreducible represen-
tations of SU (q − 1) contained in an irreducible representation of SU (q), which, for
the present case, states that the irreducible representations of SU (3), contained in the
irreducible representation of SU (4) that corresponds to a Young tableau [n1, n2, n3, n4],
n1 ≥ n2 ≥ n3 ≥ n4, are the irreducible representations of SU (3) that correspond to
all the Young tableaux [m1, m2, m3], such that n1 ≥ m1 ≥ n2 ≥ m2 ≥ n3 ≥ m3 ≥ n4.
This general rule is the basis for the Gelfand-Tsetlin patterns that can be used to label
the basis vectors of the irreducible representations of SU (q), via the subgroup chain
U (1) ⊂ SU (2) ⊂ SU (3) ⊂ . . . ⊂ SU (q − 1) ⊂ SU (q), as reviewed, for example, in
[357].
We next note that if the SU (3) representation, corresponding to a Young tableau
[m1, m2, m3], is contained in an SU (4) representation, corresponding to a Young tableau
[n1, n2, n3, n4], with n4 ≥ 0, then m1 +m2 +m3 of the n1 + n2 + n3 + n4 copies of the
SU (4) fundamental, from which the SU (4) representation is constructed, branch to
the 3 of SU (3), and the remaining n1 + n2 + n3 + n4 − (m1 +m2 +m3) copies of the
SU (4) fundamental branch to the 1 of SU (3), so from (455), the U (1) charge of the
SU (3) representation is
m1 +m2 +m3 −
(n1 + n2 + n3 + n4) (460)
Furthermore, this relation, like the rule for the SU (3) irreducible representations con-
tained within a given SU (4) irreducible representation, is unaltered by adding a con-
stant to all the ni and all the mi, so it remains true when the assumption that n4 ≥ 0
is no longer satisfied.
We now find that the rule (460), in combination with the rule that n1 ≥ m1 ≥
n2 ≥ m2 ≥ n3 ≥ m3 ≥ n4, is very restrictive. For example, the SU (3) representation,
corresponding to the Young tableau [0, 0, 0], is contained in all the SU (4) representa-
tions [p, 0, 0,−p′], with p ≥ 0 and p′ ≥ 0, but the SU (3)×U (1) representation [0, 0, 0]0
is contained only in the representations [p, 0, 0,−p], with p ≥ 0, whose representation
matrices are given in (459). The Dynkin labels of these representations of SU (4) are
(p, 0, p), as already noted, and in general, the Dynkin label of an SU (4) irreducible
representation, that corresponds to a Young tableau [a, b, c, d], is (a− b, b− c, c− d),
with the three components of the Dynkin label corresponding to the three vertices of
the SU (4) Dynkin diagram, written in sequence from end to end along the line.
We next note that each massive d = 4 graviton mode hµν , corresponding to har-
monics on CP3 with Dynkin labels (p, 0, p), p > 0, will absorb a d = 4 vector with
the same Dynkin label, for which hµy is available, and a d = 4 scalar with the same
Dynkin label, for which hyy is available. Many of the other d = 4 massive modes, with
a subscript y, also get absorbed by higher spin d = 4 massive modes, with matching
Dynkin labels, in a similar way, although for the modes Cµνr, Cµyr, Cµνr̄, and Cµyr̄, the
situation is reversed, with the d = 4 vector fields Cµyr and Cµyr̄ absorbing the fields
Cµνr and Cµνr̄, which are equivalent to d = 4 scalar fields. In this way, we find that the
unabsorbed d = 4 massive modes, and the SU (4) irreducible representations that occur
in their harmonic expansions on CP3, are as shown in Table 4 for the bosons, and in
Table 5 for the fermions. Most of the SU (4) irreducible representations that occur in
the harmonic expansions of the metric, and of a vector field, were listed by Sobczyk,
[344], in the context of a CP3 compactification of a d = 10 Einstein-Yang-Mills theory.
We see that for each d = 4 spin, the SU (4) multiplets in Tables 4 and 5 are in one
to one correspondence with the SU (4) multiplets listed by Nilsson and Pope [348] for
that d = 4 spin, except that in some cases, p has to be shifted by a small number.
This was to be expected, because simply listing the harmonics corresponding to each
d = 4 state does not determine the corresponding masses, nor does it determine how
the states are organized into N = 6 supermultiplets. I have listed the SU (4) multiplets
in Tables 4 and 5 so that the complete set of harmonics entering the expansion, on
CP3, of the corresponding d = 4 state, is given by the SU (4) multiplets shown, for
all p ≥ 0, whereas p, in Nilsson and Pope’s Table 1, identifies the distinct N =
6 supermultiplets, with p = 0 corresponding to the standard N = 6 supergravity
multiplet. Thus in Nilsson and Pope’s Table 1, many of the SU (4) Dynkin labels have
a negative component for small values of p, and in particular, for p = 0, indicating
the absence of an SU (4) representation in that p-series, in the corresponding low-lying
N = 6 supermultiplet.
Nilsson and Pope also listed the parities of the d = 4 boson states. I have not
calculated the parities of the d = 4 boson states by the Salam-Strathdee method, but
we note that if we assume that the parity of a d = 4 boson state is the product of
a factor of −1 if the state arises from the d = 11 three-form gauge field, a factor of
d = 11
compo-
nents
SU(3)×U(1)
tableau
SU (4) Dynkin labels
p ≥ 0
2 hµν [0, 0, 0]0 (p, 0, p)
1 hµr [1, 0, 0]1 (p+ 1, 0, p+ 1) + (p, 1, p+ 2)
1 hµr̄ [0, 0,−1]−1 (p+ 1, 0, p+ 1) + (p+ 2, 1, p)
0 hrs̄
[1, 0,−1]0
+ [0, 0, 0]0
(p+ 1, 0, p+ 1) + (p, 1, p+ 2) + (p+ 2, 1, p)
+ (p, 2, p) + (p, 0, p)
0 hrs [2, 0, 0]2 (p+2, 0, p+2)+ (p+1, 1, p+3) + (p, 2, p+4)
0 hr̄s̄ [0, 0,−2]−2 (p+2, 0, p+2)+ (p+3, 1, p+1) + (p+4, 2, p)
0 Cµνy [0, 0, 0]0 (p, 0, p)
1 Cµyr [1, 0, 0]1 (p+ 1, 0, p+ 1) + (p, 1, p+ 2)
1 Cµyr̄ [0, 0,−1]−1 (p+ 1, 0, p+ 1) + (p+ 2, 1, p)
1 Cµrs̄
[1, 0,−1]0
+ [0, 0, 0]0
(p+ 1, 0, p+ 1) + (p, 1, p+ 2) + (p+ 2, 1, p)
+ (p, 2, p) + (p, 0, p)
1 Cµrs [0, 0,−1]2 (p, 1, p+ 2) + (p, 0, p+ 4)
1 Cµr̄s̄ [1, 0, 0]−2 (p+ 2, 1, p) + (p+ 4, 0, p)
0 Crst [0, 0, 0]3 (p, 0, p+ 4)
0 Crst̄
[0, 0,−2]1
+ [1, 0, 0]1
(p, 0, p+ 4) + (p, 1, p+ 2) + (p, 2, p)
+ (p+ 1, 0, p+ 1) + (p, 1, p+ 2)
0 Crs̄t̄
[2, 0, 0]−1
+ [0, 0,−1]−1
(p+ 4, 0, p) + (p+ 2, 1, p) + (p, 2, p)
+ (p+ 1, 0, p+ 1) + (p+ 2, 1, p)
0 Cr̄s̄t̄ [0, 0, 0, ]−3 (p+ 4, 0, p)
Table 4: Boson harmonics for type IIA d = 10 supergravity compactified on CP3.
d = 4
d = 11
compo-
nents
SU(3)×U(1)
tableau
SU (4) Dynkin labels
p ≥ 0
ψµ [0, 0, 0]− 3
(p+ 2, 0, p)
ψµ [0, 0,−1] 1
(p, 0, p+ 2) + (p, 1, p)
ψµ [0, 0, 0]3
(p, 0, p+ 2)
ψµ [1, 0, 0]− 1
(p+ 2, 0, p) + (p, 1, p)
ψr [1, 0, 0]− 1
(p+ 2, 0, p) + (p, 1, p)
ψr [1, 0,−1] 3
(p+1, 0, p+3)+ (p, 1, p+4) + (p+1, 1, p+1)
+ (p, 2, p+ 2)
ψr [0, 0, 0]3
(p, 0, p+ 2)
ψr [1, 0, 0]5
(p+ 1, 0, p+ 3) + (p, 1, p+ 4)
ψr [2, 0, 0]1
(p+ 2, 0, p) + (p+ 1, 1, p+ 1) + (p, 2, p+ 2)
ψr [0, 0,−1] 1
(p, 0, p+ 2) + (p, 1, p)
ψr̄ [0, 0,−1]− 5
(p+ 3, 0, p+ 1) + (p+ 4, 1, p)
ψr̄ [0, 0,−2]− 1
(p, 0, p+ 2) + (p+ 1, 1, p+ 1) + (p+ 2, 2, p)
ψr̄ [1, 0, 0]− 1
(p+ 2, 0, p) + (p, 1, p)
ψr̄ [0, 0,−1] 1
(p, 0, p+ 2) + (p, 1, p)
ψr̄ [1, 0,−1]− 3
(p+3, 0, p+1)+ (p+4, 1, p) + (p+1, 1, p+1)
+ (p+ 2, 2, p)
ψr̄ [0, 0, 0]− 3
(p+ 2, 0, p)
Table 5: Fermion harmonics for type IIA d = 10 supergravity compactified on CP3.
−1 if the d = 4 boson state has spin 1, a factor of −1 if the second component of
the SU (4) Dynkin label is an odd number, and a factor of −1 for each index y of the
d = 11 components that the state arises from, then the SU (4) boson multiplets listed
in Table 4, for each d = 4 spin and parity, can be paired one to one with the SU (4)
boson multiplets listed by Nilsson and Pope, of the same d = 4 spin and parity, up to
small shifts of p in some cases, as before.
We can now calculate the one-loop vacuum energy of the Watamura N = 6 com-
pactification of type IIA d = 10 supergravity, by the method of Gibbons and Nicolai
[340], which uses the zeta function regularization method of Hawking [358]. We can
directly use Gibbons and Nicolai’s formula (9) for the contribution to the vacuum en-
ergy from a spin s state, s > 0, and their formula (11) for the contribution to the
vacuum energy from spin 0 state, except that the last term in their formula (9), in
the scanned version of the preprint from KEK [340], which seems to be a misprint,
has to be replaced by −
20s+9
. I have verified, using Maxima [291], that the one-
loop vacuum energy of the Freund-Rubin compactification does, indeed, vanish floor
by floor, for all Kaluza-Klein levels above the lowest, when this replacement is made
in their formula (9). The lowest Kaluza-Klein level, namely the ordinary supergravity
multiplet, requires a separate calculation, which was carried out by Allen and Davis
[339], because the formula for the dimension of an SO (8) irreducible representation
does not vanish for some of the SO (8) Dynkin labels with a negative component that
arise in this case, such as (−2, 1, 0, 0).
The dimensions of the irreducible representations of SU (4) with Dynkin labels
(a, b, c), where the integers a ≥ 0, b ≥ 0, and c ≥ 0 are associated with the three
verteices of the SU (4) Dynkin diagram, taken in sequence from end to end along
the line, can be calculated from Weyl’s dimension formula [359, 360], or from the
combinatorial result, summarized in section 3.I.(b) of [361], that the dimension of the
irreducible representation of GL (n,C) associated with an ordinary Young tableau with
n rows, and no negative length rows, is the product over all the boxes x of the tableau,
n+c(x)
, where c (x) is the horizontal position of x minus its vertical position, counting
from left to right and downwards, starting from the box at the top left-hand corner
of the tableau, and hx, the length of the hook whose top left-hand corner is x, is the
number of boxes directly under x, plus the number of boxes directly to the right of x,
plus 1. The result is:
D (a, b, c) = (1 + a) (1 + b) (1 + c)
a+ b+ c
(461)
Then using Maxima [291], we find, by a formula analogous to Gibbons and Nicolai’s
formula (14), with z = −1, but using the entries in Nilsson and Pope’s Table 1,
instead of from Gibbons and Nicolai’s Table 1, that in units of
, where Λ is
the cosmological constant of the AdS4, the contributions to the vacuum energy, from
the states in the N = 6 supersymmetry multiplet at Kaluza-Klein level p, for p ≥ 2,
where p = 0 corresponds to the N = 6 supergravity multiplet, are as follows.
Spin 2:
−20p9 − 270p8 − 1580p7 − 5250p6 − 10888p5 − 14565p4 − 12506p3
−6597p2 − 1916p− 228
(462)
Spin 3
320p9 + 4320p8 + 24800p7 + 78960p6 + 152188p5 + 181290p4 + 129622p3
+50049p2 + 7275p− 414
(463)
Spin 1:
−180p9 − 2430p8 − 13740p7 − 42210p6 − 76152p5 − 80685p4 − 45930p3
−9045p2 + 2892p+ 1260
(464)
Spin 1
960p9 + 12960p8 + 72480p7 + 216720p6 + 370644p5 + 353070p4
+153770p3 − 7065p2 − 28859p− 6690
(465)
Spin 0:
−420p9 − 5670p8 − 31500p7 − 92610p6 − 152928p5 − 134865p4 − 45442p3
+14211p2 + 13360p+ 2004
(466)
The fact that the contribution of the spin 2 states is negative is presumably an artifact
of the zeta function regularization used. The sum of these contributions is zero, so for
all the Kaluza-Klein levels with p ≥ 2, the one-loop vacuum energy vanishes “floor by
floor” for the Watamura-Nilsson-Pope N = 6 CP3 compactification of type IIA d = 10
supergravity, just as it does for the N = 8 Freund-Rubin compactification of d = 11
supergravity.
The cases of p = 0 and p = 1 require separate calculations, because some SU (4)
multiplets occur that should be omitted in these cases, and some Dynkin labels with a
negative component occur, for which the formula (461) for the dimension of an SU (4)
irreducible representation does not give zero. For the N = 6 supergravity multiplet,
which is the case with p = 0, the one-loop result vacuum energy was found to vanish
by Allen and Davis [339]. The case of p = 1 requires further study, and will not be
considered in this paper. But it does not seem very likely that the one-loop vacuum
energy would fail to vanish for this one Kaluza-Klein level, when it does for all the
others.
We note that this calculation has not included any Kaluza-Klein excitations asso-
ciated with the S1, so in the context of Nilsson and Pope’s compactifications of d = 11
supergravity, interpolating between the Freund-Rubin compactification of d = 11 su-
pergravity, and Watamura’s N = 6 compactification of type IIA d = 10 supergravity,
the d = 4 states listed here are appropriate for the limit in which the radius of the
S1 tends to zero. To consider the opposite limit, in which the radius of the S1 tends
to infinity, which is presumably related to the N = 6 supergravity in five dimensions
listed by Cremmer [362], it would be necessary to repeat the calculation with the ex-
tra modes included. However, at a certain value of the radius of the S1, the N = 6
supersymmetry would be extended to the N = 8 supersymmetry of the Freund-Rubin
compactification, for which it is known from the Gibbons-Nicolai calculation that the
one-loop vacuum energy vanishes floor by floor. So it is perhaps plausible that the
one-loop vacuum energy might also vanish floor by floor for all values of the radius of
the S1, from 0 to ∞.
If the numbers of fermion and boson helicity states are equal for all the N = 6
massive multiplets, which I have not explicitly checked, then we would presumably
find that the one-loop vacuum energy would still vanish when the background is flat
four-dimensional Minkowski space times CP3, even though this background is not a
solution of the classical field equations, and is not supersymmetric, and, on the basis
of relations between the propagators and heat kernels on Minkowski space times CP3,
and on Minkowski space times CH3, it might then also vanish when the background
is flat four-dimensional Minkowski space times CH3. And for similar reasons, it seems
possible that the one-loop vacuum energy might also vanish when the background is
flat five-dimensional Minkowski space times CH3.
However, in consequence of the rule, discussed at the beginning of this section, that
the quantum effective action of the BRST-BV gauge-fixed theory, in a background that
is not a solution of the classical field equations, is the sum of all the one-line-irreducible
vacuum bubbles, calculated with an action given by the BRST-BV gauge-fixed classical
action in the presence of the background field, but with the terms linear in the quantum
fields deleted, the action used in the calculation of the quantum effective action of
the BRST-BV gauge-fixed theory, on a background that is flat four-dimensional or
five-dimensional Minkowski space, times CP3 or CH3, would presumably not satisfy
identities needed to use Zumino’s arguments [338] for the vanishing of the higher loop
vacuum energies.
Nevertheless, if the massive N = 6 multiplets satisfied Curtright’s spin sum rules
[363] for a theory with N = 6 supersymmetry in d = 4, which I have not explicitly
checked, then some of the ingredients for a possible cancellation of higher loop vacuum
energies, on a flat four-dimensional Minkowski space times CP3 or CH3 background,
would be in place, so the possibility that such cancellations might occur is not yet
excluded. However, the grounds for expecting such higher loop cancellations to occur
are not very strong, and it does not seem very likely that the higher loop vacuum
energies of type IIA d = 10 supergravity on a four-dimensional Minkowski space times
uncompactified CH3 background, and of d = 11 supergravity on a five-dimensional
Minkowski space times uncompactified CH3 background, will vanish, notwithstanding
the special properties of the Watamura N = 6 compactification of type IIA d = 10
supergravity, and its oxidation to d = 11 by Nilsson and Pope, for three reasons.
Firstly, the lowest Kaluza-Klein energies of the states in the N = 6 supermultiplet
at Kaluza-Klein level p, are not all the same. Instead, they differ by up to four units
within the same multiplet, so the energy differences, between the lowest
energies of states within one multiplet, are up to four times greater than the energy
difference between corresponding states within successive multiplets, and these energy
differences, between the states within a multiplet, are likely to be essential for the
cancellation of the vacuum energy of a multiplet, at least when Λ is nonzero.
The contribution to the vacuum energy, from a state of lowest energy E0, is a
quartic polynomial in E0, by Gibbons and Nicolai’s equations (9) and (11). E0 is
linear in the Kaluza-Klein level number p. When the relation between the sectional
curvature of the AdS4, and the minimum sectional curvature of the CP
3, is broken,
there are two independent units of energy, namely
, where
ΛCP3 is defined in terms of the minimum sectional curvature of the CP
3. We would
now expect the lowest energy E0 of a state to contain a term p
, associated
with its Kaluza-Klein level number p, and a term q
, where q is the integer or
half integer, such that 1 ≤ q ≤ 5, that determines the offset of E0 from p
listed by Nilsson and Pope, for the case when ΛCP3 = Λ.
Vanishing of the one-loop vacuum energy floor by floor, for independent Λ and
ΛCP3 , would then require that the coefficients of the different powers of
in the vacuum energy, which are polynomials in p of degree up to 9, all
vanish separately, and, although this has not been excluded, there is no reason to expect
it to happen, to the best of my knowledge, except that, if the numbers of fermion and
boson states in each N = 6 multiplet are equal, we would expect the coefficients of the
terms independent of Λ, which are polynomials in p of degree 9, to vanish, since the
one-loop vacuum energy of each N = 6 multiplet would then vanish in four-dimensional
Minkowski space.
Thus, although the one-loop vacuum energy of each N = 6 multiplet would vanish
for Λ = 0, if the numbers of fermion and boson states in each N = 6 multiplet are equal,
it does not seem very likely that the one-loop vacuum energy of each N = 6 multiplet
would vanish for values of the ratio Λ
strictly between 0 and 1, so the vanishing
of the one-loop vacuum energy of each N = 6 multiplet, for Λ = 0, would be an
isolated phenomenon, not continuously connected to the supersymmetric system with
Λ = ΛCP3 , so it does not seem very likely that the supersymmetric system could result
in the vanishing of the higher loop vacuum energies of type IIA d = 10 supergravity on a
four-dimensional Minkowski space times CP3 background, or that its d = 11 oxidation
could result in the vanishing of the higher loop vacuum energies of d = 11 supergravity
on a five-dimensional Minkowski space times CP3 background, when there are no other
reasons to expect this to happen.
Secondly, there is a second Watamura-Nilsson-Pope CP3 compactification of type
IIA d = 10 supergravity, that has no supersymmetry, but differs from the N = 6 com-
pactification only by the relative orientation of a four-form field strength Fµνστ , which
is proportional to the d = 4 tensor density ǫµνστ , and a two-form field strength FAB,
which is proportional to the Kähler form of the CP3. The relative orientation of Fµνστ
and FAB is detected by the supersymmetry variations of the fermions. Now, by the
Salam-Strathdee construction, the small fluctuation modes, about this N = 0 compact-
ification, will consist of exactly the same collection of series of SU (4) representations
as listed above for the N = 6 compactification, but the lowest energies, of the smallest
representations of some of the series, will be shifted up or down, by a small number of
units of
, so that the vacuum energy will presumably no longer vanish.
And to distinguish the two cases, both Fµνστ and FAB would still have to be nonzero,
when the AdS4 is replaced by Minkowski space, and the two cases would still have to be
distinguished, when the CP3 is replaced by CH3, so it seems unlikely that the higher
loop vacuum energies will vanish for a four-dimensional Minkowski space times CH3
background, without nonvanishing background fields corresponding to Fµνστ and FAB.
And for the corresponding compactifications of d = 11 supergravity, Nilsson and Pope
showed that these twoCP3 compactifications of type IIA d = 10 supergravity are “Hopf
fibrations” of the Freund-Rubin AdS4 × S7 compactification of d = 11 supergravity,
which means that the metric ansatz (94) would have to be modified by the replacement
(454), where AA is proportional to a potential for the Kähler form of the CP
3 or CH3.
And thirdly, there is an AdS5 × CP3 compactification of d = 11 supergravity,
such that the only nonvanishing form field, in the background, has the form of the
Lukas-Ovrut-Stelle-Waldram [68] ansatz (147). This compactification is investigated
in the following subsection 4.2, and found to have no supersymmetry. Its one-loop
vacuum energy will thus presumably be nonvanishing, and d = 11 supergravity, on a
five-dimensional Minkowski space times CP3 or uncompactified CH3 background, is as
closely related to this compactification, as it is to Nilsson and Pope’s d = 11 oxidation
of Watamura’s N = 6 compactification of type IIA d = 10 supergravity. This suggests,
again, that if the higher loop vacuum energies of d = 11 supergravity were to vanish
on any five-dimensional Minkowski space times CH3 background, there would have
to be a nonvanishing field strength Fµνστ in the background, and the metric ansatz
(94) would have to be modified by the replacement (454), with AA proportional to a
potential for the Kähler form of the CH3, in order to relate the background to the
d = 11 oxidation of the Watamura N = 6 compactification, and distinguish it from a
background related to the AdS5 ×CP3 compactification.
These arguments do not exclude the possibility that the higher loop vacuum energies
of a four-dimensional Minkowski space times uncompactifiedCH3 timesR1 background
for d = 11 supergravity, with suitable dependences of a (y) and b (y), in the metric
ansatz (94), on the position y along the R1, and a suitable y-dependent value of the
field strength Fµνστ , proportional to ǫµνστ , and the replacement (454) in the metric
ansatz (94), with AA a suitable y-dependent multiple of a potential for the Kähler
form of the CH3, might vanish. However, the reasons for expecting such a background
to exist, for which the higher loop vacuum energies vanish, are not very strong, so for
the phenomenological estimates in this paper, I assume that the higher-loop vacuum
energies are nonvanishing on an uncompactified CH3 background, and, moreover, that
they have their typical order of magnitude, in terms of the magnitude of the curvature
of the background, which means that b1
cannot be smaller than the value ∼ 0.03 to
0.2 estimated in subsection 2.3.6, on page 66, on the basis of Giudice, Rattazzi, and
Wells’s estimate [11] of the expansion parameter for quantum gravitational corrections
in d dimensions, so that, in consequence of the relation (103), on page 37, between
and |χ (M6)|, which follows from the estimate (102), of the d = 4 Yang-Mills fine
structure constant at unification, values of |χ (M6)| larger than ∼ 7 × 104 to 6 × 109
are excluded.
If it turned out that cancellations of higher loop vacuum energies of Hořava-Witten
theory, on a suitable uncompactified CH3 background, actually did occur, allowing
to be smaller than ∼ 0.03 to 0.2, and |χ (M6)| to be larger than ∼ 7 × 104 to
6 × 109, when M6 is a smooth compact quotient of CH3 that is a spin manifold,
then the phenomenological estimates in this paper would presumably still be valid,
with minor modifications, for smooth compact quotients of H6 that are spin manifolds,
since, to the best of my knowledge, there is no reason to expect the vacuum energy of
Hořava-Witten theory to vanish on an uncompactified H6 background.
It would be interesting to find out whether Nilsson and Pope’s d = 11 “oxidation”
of the Watamura N = 6 CP3 compactification of type IIA d = 10 supergravity,
as discussed in this subsection, can be extended by the addition of non-vanishing
components GABCD of the four-form field strength of the three-form gauge field, given
by the ansatz (147) of Lukas, Ovrut, Stelle, and Waldram (LOSW) [68], so as to obtain
a supersymmetric AdS4 × CP3 compactification of Hořava-Witten theory, consistent
with Witten’s topological constraint [45], when the SU (3) part of the spin connection
of the CP3 is embedded in the E8 on one of the two orbifold hyperplanes, and the
U (1) part of the spin connection of the CP3 is embedded in the E8×E6 left unbroken
by the SU (3) embedding, in one of the four ways listed by Pilch and Schellekens, in
subsection 4.3 of [268].
However the components Gµνστ of the four-form field strength of the three-form
gauge field, like the components GABCD, are odd under reflection in the Hořava-Witten
orbifold hyperplanes, so if they do not vanish as one or both of the orbifold hyperplanes
are approached, they would have to have discontinuities at the orbifold hyperplanes in
the upstairs picture, which would then, by (43), require the existence of non-vanishing
components Fµν of the E8 Yang-Mills field strength on the corresponding orbifold
hyperplane, which would break invariance under the SO (3, 2) Anti de Sitter group.
Thus to preserve invariance under the Anti de Sitter group, Gµνστ would have to
vanish on both orbifold hyperplanes. This is not necessarily inconsistent with the
existence of a compactification, since there also exists an AdS5×CP3 compactification
of d = 11 supergravity, whose only non-vanishing components of GIJKL are given by
the LOSW ansatz (147), but I shall show in the next subsection that this AdS5×CP3
compactification has no supersymmetry, so to have a chance of having a supersymmetric
AdS4×CP3 compactification of Hořava-Witten theory, Gµνστ would have to be nonzero
in the bulk, away from the orbifold hyperplanes. The boundary conditions, on Gµνστ ,
would then be that these components vanish on both orbifold hyperplanes. A new
feature, in the bulk, would be that GIJKL now has enough nonvanishing components,
in the bulk, to turn on the nonlinear GIJKL1...L8G
L1...L4GL5...L8 term in the field equation
for CIJK, where GIJKL1...L8 denotes the tensor
−GǫIJKL1...L8. We would thus expect
also to find some nonvanishing components of GIJKL that have an index y, and some
nonvanishing components of CIJK , with an index y, were in fact found in Witten’s
original investigation of supersymmetric compactifications of Hořava-Witten theory
[127].
Furthermore, the metric components GAy, which are nonzero in the Nilsson-Pope
d = 11 oxidation of the Watamura N = 6 compactification of type IIA d = 10 su-
pergravity, due to the replacement (454) in the metric ansatz (94), are also odd under
reflection in the Hořava-Witten orbifold hyperplanes, and must thus presumably vanish
on the orbifold hyperplanes, since, to the best of my knowledge, there is no analogue,
for the metric components GUy, of the discontinuity equation (43), for the components
GUVWX of the four-form field strength of the three-form gauge field. This is, again, not
necessarily inconsistent with the existence of a compactification, due to the existence
of the AdS5×CP3 compactification studied in the next subsection, and would give the
boundary conditions on GAy.
In the presence of a boundary, half of the bulk supersymmetry is always broken
[364]. However the N = 3 d = 4 supergravity supermultiplet contains three vector
bosons, which naturally transform as the adjoint of SO (3), and do not fit naturally
into aCP3 compactification. However, as noted by Nilsson and Pope [348], the SU (4)×
U (1) gauge bosons, found in the Watamura CP3 compactification of type IIA d = 10
supergravity, could be consistent with N = 2 or N = 1 supersymmetry, as well as
with N = 6 supersymmetry. N = 2, d = 4 supersymmetry is not consistent with the
existence of chiral fermions, and three of the four embeddings of the SU (3) × U (1)
spin connection of CP3, in E8×E8, found by Pilch and Schellekens [268], have chiral
fermions, so could have at most N = 1 supersymmetry, whereas the fourth embedding
found by Pilch and Schellekens, their case 4.3.(a), has no chiral fermions for CP3, and
thus might possibly be consistent with N = 2 supersymmetry.
We note that for gauged N -extended d = 4 supergravity, with Nleq4, and not cou-
pled to any matter multiplets, Allen and Davis [339] found that the one-loop vacuum
energy, in the AdS4 background, is nonvanishing, so that there would be no possibility
of an analogue of the Gibbons-Nicolai floor by floor vanishing of the one-loop vacuum
energy when the contributions of the Kaluza-Klein multiplets above the supergravity
multiplet are included. However, if the Nilsson-Pope d = 11 oxidation of the Watamura
N = 6 solution could be modified to obtain a supersymmetric CP3 compactification
of Hořava-Witten theory, in the manner just discussed, there would be additional su-
persymmetric Yang-Mills multiplets, together with the Kaluza-Klein multiplets above
them, so there would be a possibility that the floor by floor vanishing of the one-loop
vacuum energy might be restored.
The question of whether or not there exists, in the bulk, a supersymmetric deforma-
tion of the Nilsson-Pope d = 11 oxidation of the Watamura N = 6 compactification,
whose nonvanishing components of GIJKL include components GABCD given by the
LOSW ansatz (147), where α might now depend on y, could perhaps be investigated,
in the first instance, by Witten’s method [127], in which the new components of GIJKL
would be treated as a perturbation.
4.2 AdS5 ×CP3 compactification of d = 11 supergravity
The value of the integration constant B, in (206), that is required for TeV-scale gravity,
is given by (313), when the outer surface of the thick pipe is stabilized in the quantum
region by Casimir effects, and by (399), when the outer surface is stabilized in the
classical region by extra fluxes. From these equations, we see that the value of B
required for TeV-scale gravity is reduced if the Euler number χ (M6) of the compact
six-manifold, which is a negative integer for the compact six-manifolds considered in
the present paper, is large in magnitude. However, |χ (M6)| is also related to b1
the relation (103), which follows from the value (102) of the Yang-Mills fine structure
constant assumed at unification, which is the value of the QCD fine structure constant,
α3, as evolved in the Standard Model to around 150 TeV. Here b1 = b (y1) is the
value of b (y) at the inner surface of the thick pipe, where b (y) was introduced in the
metric ansatz (94) as the scale factor that determines the diameter of the compact
six-manifold, once its topology is fixed by selecting a specific smooth compact quotient
of CH3 or H6. And b1
is determined by Casimir effects near the inner surface of the
thick pipe, and thus, as discussed in subsection 2.4.3, cannot be small compared to 1,
unless, for some reason, not only are the one-loop coefficients in the Casimir energy
densities (132) and (136) small compared to 1, but also the multi-loop coefficients are
all suppressed by the appropriate powers of the small number b1
, either to all loop
orders, or at least up to some high loop order. Thus we cannot have a very large value
of |χ (M6)|, and also obtain a reasonable value value of the Yang-Mills fine structure
constant at unification, unless the coefficients in the Casimir energy densities (132) and
(136), either to all loop orders, or at least to some high loop order, all tend to zero
as the appropriate power of b1
, where b1
is given by (103), as |χ (M6)| becomes
very large. However there is no reason for this to happen unless some special effect
occurs, because the limit |χ (M6)| → ∞ does not correspond to any restoration of
supersymmetry.
A special effect of the required type was, however, discovered by Gibbons and
Nicolai [340], who calculated the one-loop Casimir energy density of the Freund-Rubin
compactification of d = 11 supergravity on the round seven-sphere [346], including the
effects of all the Kaluza-Klein states, and found that not only did the Casimir energy
density vanish, as required to preserve the supersymmetry of the solution at one loop,
but also the contributions to the Casimir energy density vanished “floor by floor”, or
in other words, at each separate Kaluza-Klein level or Osp(8|4) multiplet, which is not
required to preserve the supersymmetry. This appears to suggest that the one-loop
Casimir energy density of this compactification would still vanish “floor by floor” even
if the Freund-Rubin relation between the AdS4 radius and the S
7 radius was broken, in
which case the background would no longer satisfy the classical Cremmer-Julia-Scherk
field equations, but the Casimir energy density would nevertheless still be defined by
the general formula for the quantum effective action, Γ, as a function of arbitrary
background fields, as described before (129). Thus the Gibbons-Nicolai result would
seem to imply that the one-loop Casimir energy density of d = 11 supergravity, defined
in this way, would vanish “floor by floor” even when the background is flat R4, times
S7. And furthermore, since there will be relations between the propagators and heat
kernels on a flat R4, times S7, background, and the propagators and heat kernels on a
flatR4, timesH7, background, analogous to those discussed above for the flatR5, times
CP3, and the flat R5, times CH3, backgrounds, the Gibbons-Nicolai result would seem
to suggest that the one-loop Casimir energy density of d = 11 supergravity, defined by
the quantum effective action, Γ, as a function of arbitrary background fields, will also
vanish when the background is flat R4, times H7, for arbitrary radius of curvature of
the H7. In that case, the one-loop Casimir energy density of d = 11 supergravity, on a
flat R4, times M7, background, where M7 is a smooth compact quotient of H7, would
presumably tend to zero, in the limit as the volume of the M7 at fixed Ricci scalar,
which is a topological invariant by Mostow’s rigidity theorem even though the Euler
number vanishes for a smooth compact manifold of odd dimension, tends to infinity.
Thus it is appropriate to ask if there exist supersymmetric compactifications of
d = 11 supergravity on AdS5 ×CP3 or AdS5 × S6, which might lead, by an analogue
of the Gibbons-Nicolai effect, to the vanishing of the one-loop Casimir energy density
of d = 11 supergravity, as defined by the quantum effective action, Γ, on a flat R5,
times CH3, background, or a flat R5, times H6, background. To the best of my
knowledge, there is no classical solution of the Cremmer-Julia-Scherk field equations
on an AdS5×S6 background, that has a maximally symmetric metric on both factors,
because there is no natural ansatz for the four-form field strength of the three-form
gauge field. However, there is, indeed, a classical solution of the Cremmer-Julia-Scherk
field equations on an AdS5 ×CP3 background, with the Lukas-Ovrut-Stelle-Waldram
(LOSW) ansatz (147) for the four-form field strength of the three-form gauge field. I
shall seek a solution with the metric ansatz (94), such that AdS5 is realized as flat four-
dimensional Minkowski space times the y direction, with a (y) depending exponentially
on y, as in the Randall-Sundrum model [31], and b (y) independent of y. Comparing the
Ricci tensor components (97), the energy-momentum tensor components contributed
by the three-form gauge field with the LOSW ansatz (159), the definition of the t(i) (y)
energy-momentum tensor coefficients (130), and the Einstein equations (162), (163),
and (164), we see that on replacing CH3 by CP3, so that the relation RAB (h) = 4hAB
is replaced by RAB (h) = −4hAB, and replacing four-dimensional de Sitter space by
four dimensional Minkowski space, so that the relation Rµν (g) = −3gµν is replaced by
Rµν (g) = 0, and setting the t
(i) (y) energy-momentum tensor coefficients to the values
given by the LOSW ansatz, the Einstein equations become:
= 0 (467)
= 0 (468)
= 0 (469)
Requiring that ḃ = 0, b̈ = 0, the second of these equations reduces to
α2 = 36b6 (470)
The first and third equations then reduce to:
(471)
(472)
which have the solutions a = Ae
b and a = Ae−
b . And from the formulae (96) for the
Riemann tensor components, we see that
Rµνστ =
(GµσGντ −GνσGµτ ) , Rµyνy =
Gµν (473)
hence since Gµy = 0 and Gyy = 1, we have:
Rµ̄ν̄σ̄τ̄ =
(Gµ̄σ̄Gν̄τ̄ −Gν̄σ̄Gµ̄τ̄ ) (474)
where the barred Greek indices run over four-dimensional Minkowski space and y. Thus
the five-dimensional space formed from four-dimensional Minkowski space and the y
direction is maximally symmetric, and in consequence of its (−++++) signature
and the relation Rµ̄σ̄ =
Gµ̄σ̄, is AdS5.
We now need to determine whether this solution has any supersymmetries. There
are no Majorana spinors in five dimensions, but a symplectic-Majorana condition can
be imposed on a pair of spinors [362], in consequence of which the possible numbers
of supersymmetries in five dimensions are even, and there do, indeed, exist supergrav-
ities with 2, 4, 6, and 8 supersymmetries in five dimensions [362]. We know from the
Figueroa-O’Farrill - Papadopoulos theorem [365] that the solution cannot have 8 super-
symmetries. The SU(4) isometry group of CP3 with its standard metric implies there
will be 15 Yang-Mills vector bosons in the adjoint of SU(4), and looking at the table
of states of the extended supergravities in five dimensions given by Cremmer [362], we
see that N = 6 supergravity in five dimensions has precisely 15 vector fields, which
on toroidal compactification to four dimensions join the extra vector field coming from
the metric, to produce the standard 15+ 1 = 16 vector fields of N = 6 supergravity in
four dimensions. Furthermore, Nilsson and Pope [348] found that a known compact-
ification [347] of Type IIA supergravity in ten dimensions on AdS4 × CP3 has either
N = 6 supersymmetry or no supersymmetry, depending on the relative sign of form
field fluxes on the AdS4 and CP
3 factors. However, notwithstanding these positive
indications, the AdS5×CP3 compactification of d = 11 supergravity considered above
has no supersymmetry.
To check this, I shall use the notations of subsection 2.1 for supergravity in eleven
dimensions, so coordinate indices I, J,K, . . . run over all directions on M11. The Dirac
matrices ΓI satisfy
ΓI ,ΓJ
= 2GIJ , and ΓI1I2...In ≡ Γ[I1ΓI2 . . .Γ In]. Coordinate indices
µ, ν, σ, . . . will now run over all directions on AdS5, which is a change from the meaning
of the Greek indices used above and in section 2, and coordinate indices A,B,C will run
over all directions on the compact six-manifold, which is in agreement with section 2,
although the compact six-manifold is now CP3. Local Lorentz indices will be indicated
by putting a bar over the corresponding coordinate indices, so the meaning of barred
Greek indices is also now changed from their meaning in equation (474) above. A
real representation of the Γ matrices for eleven dimensions does not decompose neatly
into Dirac matrices for the five extended dimensions with signature (−++++) and
Dirac matrices for the six compact dimensions with signature (+ + + +++), so I shall
instead use a representation of the form used by Lukas, Ovrut, Stelle, and Waldram
[68], with ΓI = 1
γµ × λ, 1× λA
, where γµ and λA are the five- and six-dimensional
Dirac matrices, respectively. Here, λ is the chiral projection matrix in six dimensions
with λ2 = 1. For a specific representation of the λĀ we can choose λ5 = σ1 × 1 × 1,
λ6 = σ2×1×1, λ7 = σ3×σ1×1, λ8 = σ3×σ2×1, λ9 = σ3×σ3×σ1, λ10 = σ3×σ3×σ2.
We define λ = iλ5λ6λ7λ8λ9λ10 = σ3 × σ3 × σ3. For a specific representation of the γµ̄
we can choose γ1 = σ1 × 1, γ2 = σ2 × 1, γ3 = σ3 × σ1, γ4 = σ3 × σ2, γ0 = iσ3 × σ3.
Then for the charge conjugation matrix C in eleven dimensions, which satisfies as usual
C = −CΓI , CT = −C, we can take C = C5×C6, where C5 = σ1×σ2 is the charge
conjugation matrix in five dimensions, and satisfies (γµ)
C5 = C5γ
µ, CT5 = −C5, in
agreement with [362], and C6 = σ2 × σ1 × σ2 is the charge conjugation matrix in six
dimensions, and satisfies
C6 = −C6λA, λTC6 = −C6λ, CT6 = C6.
Now the gravitino field is zero in the above classical solution, so if it has any super-
symmetries, there must exist supersymmetry variation parameters η (x, z), where xµ
are coordinates on AdS5, and z
A are coordinates on CP3, such that the supersymme-
try variation of the gravitino vanishes. The supersymmetry variation of the gravitino,
about a configuration in which the gravitino field is zero, is [14, 2, 127]:
δψI = DIη +
(ΓIJKLM − 8GIJΓKLM)GJKLMη (475)
To study the condition on η that results from setting this variation equal to zero, when
GJKLM is given by the LOSW ansatz (147), I shall follow the method of Nilsson and
Pope [348]. It is convenient, first, to note the identities:
ΓIJKLMG
JKLM = (ΓIΓJKLM − 4GIJΓKLM)GJKLM (476)
8GIJΓKLMG
JKLM = [ΓI ,ΓJKLM ]G
JKLM (477)
Thus the supersymmetry variation of the gravitino, (475), can be written:
δψI = DIη +
(−ΓIΓJKLM + 3ΓJKLMΓI)GJKLMη (478)
Now from the definition (69) of the Kähler form, we have:
C = −hAC (479)
Furthermore, for an arbitrary 2n × 2n antisymmetric matrix M , with real matrix
elements, we have the identity:
εi1i2i3i4...i2n−1jMi1i2Mi3i4 . . .Mi2n−1k = 2
n−1 (n− 1) !
detMδjk (480)
This is proved by applying an orthogonal similarity transformation to transform M
to a block diagonal matrix M̃ , such that each block in the block diagonal of M̃ is an
antisymmetric 2× 2 matrix with real matrix elements, then replacing each index i by
an index pair aI, where a runs from 1 to 2, and I runs from 1 to n, so that M̃ can be
expressed as a Kronecker product M̃aI,bJ = εabM̄IJ , where εab =
, and M̄ is
an n× n diagonal matrix with real matrix elements.
Applying this to the Kähler form, we have the identity:
ωABωCDh
ABCDEF = 8ωEF (481)
Following Nilsson and Pope, it is convenient to define:
Q ≡ −iωABλABλ (482)
We note that [Q, λ] = 0. And from the definition of the ΓA in terms of the λA, as
above, {λA, λB} = 2hAB. Thus from the identities
λABλCD = λABCD−hACλBD+hADλBC+hBCλAD−hBDλAC−hAChBD+hADhBC (483)
λABCD =
hABCDEFh
EGhFHλGHλ (484)
and (481), we find that:
Q2 = 4Q+ 12 (485)
Hence the eigenvalues of Q are −2 and 6, hence since Q is traceless, there are six
eigenvalues −2 and two eigenvalues 6.
We now assume that η (x, z) factorizes in the form η (x, z) = ε (x) η̃ (z), where
ε (x) is a four component spinor acted on by the first factor in the Kronecker product
expressions for the ΓI , and η̃ (z) is an eight component spinor acted on by the second
factor in the Kronecker product expressions for the ΓI . Substituting in the LOSW
ansatz (147), and requiring that δψI = 0, we find from the components of (478) with
I along CP3 that:
DAη̃ +
3456b3
(−λAλBCDE + 3λBCDEλA)hBCDEFGωFGη̃ = 0 (486)
Now from (482) and (484), we find:
λBCDEh
BCDEFGωFG = −24Q (487)
Hence (486) reduces to:
DAη̃ ±
(λAQ− 3QλA) η̃ = 0 (488)
where I also used (470), and the sign choice corresponds to α = ±6b3. And from (482),
we also have:
{Q, λA} = −4iω BA λBλ (489)
Hence (488) is equivalent to:
DAη̃ ±
λAQ+ 3iω
A λBλ
η̃ = 0 (490)
A necessary condition for the existence of solutions of (490) is the integrability condi-
tion:
λAQ+ 3iω
A λCλ
, DB ±
λBQ+ 3iω
B λDλ
= 0 (491)
To evaluate the left-hand side of (491), we note first that with the convention (5) for the
Riemann tensor, we have [DA, DB] = −14RABCDΓ
CD = −1
R̃ABCDλ
CD, where RABCD is
the Riemann curvature of CP3 with the metric GAB = b
2hAB, and R̃ABCD =
RABCD
is the Riemann curvature of CP3 with the metric hAB. And secondly, there are no
cross terms between DA or DB, and the extra terms that came from the G
JKLM term
in (475), because the extra terms are built from the Kähler form and the vielbein,
which are covariantly constant, and the Dirac matrices with local Lorentz indices, and
λ, which are position-independent invariant tensors with respectively a vector index
and two spinor indices, and two spinor indices, and thus also covariantly constant.
To evaluate the commutator of the extra terms, we note that:
[λAQ, λBQ] = −4iω DB λAλDλQ + 4iω DA λBλDλQ+ 2λBA (4Q+ 12) (492)
[λAQ, λDλ] = −4iω CD λAλC + 2λDAλQ (493)
where (485) was used to obtain (492). Thus we find:
λAQ + 3iω
A λCλ
λBQ+ 3iω
B λDλ
ω DB λDA − ω DA λDB
ωABλQ+
λBA (Q + 6)+ω
B λCD (494)
Terms of the same structure, but with different coefficients, occurred in Nilsson and
Pope’s calculation of the corresponding commutator for the AdS4 × CP3 compactifi-
cation of Type IIA supergravity in ten dimensions [347], and in that case, for one of
two alternative choices of a relative sign, the result was that after adding the Riemann
tensor term, each nonvanishing term had a factor of (Q+ 2) at its right-hand side, so
that acting on any linear combination of the six linearly independent eigenvectors of Q
with eigenvalue −2, the commutator vanished. That does not happen in the present
case, so we have to check whether there is any further relation between the terms in
the left-hand side of (491) that might result in (491) being satisfied when acting on an
appropriate eigenvector of Q.
It is convenient now to switch to complex coordinates, as in subsection 2.2, on page
25. Barred Latin indices will now denote antiholomorphic indices, as in subsection 2.2.
Then corresponding to the Riemann tensor (72) for CHn, the Riemann tensor for CP3
R̃rs̄tū = hrs̄htū + hrūhts̄ (495)
Evaluating the left-hand side of (491) for A = r, B = s, the Riemann tensor term does
not contribute, and the result is:
λsr (10λQ+ 4Q+ 33) (496)
which is nonvanishing for any combination of λ = +1 or −1, and Q = −2 or +6, and
thus proves the absence of supersymmetry. And similarly, for A = r̄, B = s̄, the left-
hand side of (491) is 1
λs̄r̄ (−10λQ+ 4Q+ 33). And for A = r, B = s̄, the Riemann
tensor term −1
R̃ABCDλ
CD contributes 1
(iωrs̄Qλ + 2λrs̄), and the left-hand side of
(491) is 1
(25iωrs̄Qλ + λs̄r (16Q+ 42)). In fact, if the numerical coefficients of the
terms in the parentheses in (490) had had the values ±3
i, ∓3
i, instead of their actual
values 1 and 3, the left-hand side of (494) would have been equal to 1
λAB (Q+ 2), and
would thus have been consistent with N = 6 supersymmetry.
About 18 months after version 1 of this article was published on arXiv, I learned
from [366] that the AdS5 ×CP3 solution was studied by Pope and van Nieuwenhuizen
in 1989, who showed that it is not supersymmetric [367]. The lack of supersymmetry
could also have been deduced from a general study of supersymmetric AdS5 solutions
of M-theory by Gauntlett, Martelli, Sparks, and Waldram [368].
5 E8 vacuum gauge fields and the Standard Model
In the present paper, we have considered the compactification of Hořava-Witten the-
ory on a smooth compact quotient of either CH3 or H6, which breaks supersymmetry
completely. The fact that the observed gauge coupling constants are ∼ 1 in magnitude
implies that the six-volume of the inner surface of the thick pipe is ∼ κ 43 , as discussed
in subsection 2.6.1, following (309), on page 128. Thus the energy at which supersym-
metry is broken at the inner surface of the thick pipe will be ∼ κ− 29 . Thus if κ− 29 was
large compared to the energy ∼ 174 GeV at which the electroweak symmetry is broken,
we would have a hierarchy problem of the original type [369], without supersymmetry
just above the electroweak breaking energy, to stabilize the parameters of the effective
electroweak Higgs sector. Thus in models of the present type we would expect to find
the simplest physical picture if κ−
9 is as close above the electroweak breaking energy as
allowed by present experimental constraints, which in practice means TeV-scale gravity
[3, 5]. In the present section I shall consider how the Standard Model [44] might be
realized in the framework considered in the preceding sections, if κ−
9 is of order a TeV.
No positive experimental evidence for the existence of large extra dimensions and
TeV-scale gravity has yet been reported. However, in the approximation that the
seven extra dimension are flat, the branching ratio for emitting a graviton, in any
process, is ∼
κ2/9E
, where E is the energy available to the graviton [3]. Thus
if quantum gravitational effects are observed at the LHC, the effects will start very
suddenly, as the energy of the beams is gradually increased, with no detectable effects
at all up to a certain energy, and very large effects, with large amounts of missing
energy, at slightly higher beam energies, as gravitons start radiating into the bulk of
the thick pipe. This is in agreement with the general expectation that, although new
physics is not yet observed at colliders, it cannot be far away [370]. The perturbative
contributions of virtual graviton exchange to scattering amplitudes and cross sections,
not yet observed, also increase very rapidly with increasing beam energies [11, 273], and
once they become observable above the background, are expected to saturate rapidly at
the nonperturbative rate for production of short-lived microscopic black holes, whose
production cross section increases much more slowly with increasing energy, specifically
κ2/9E
[275, 371].
To estimate the current experimental limits on κ−
9 , I shall use the results of
Mirabelli, Perelstein, and Peskin [273], who consider the case of flat extra dimensions.
From the discussion around their equations (3) and (4), we see that their fundamental
gravitational mass M is defined such that for seven flat extra dimensions, compactified
to volume V7, Newton’s constant GN is given by π
16πGN
. On the other
hand, comparing (10) and (25), and remembering that for working in the “downstairs”
picture, on the manifold with boundary, the coefficient 1
in (25) is to be replaced by
, we see that V7
16πGN
. Hence κ is related to Mirabelli, Perelstein, and Peskin’s
M by 1
. Hence κ−
9 = π
≃ 0.2053M . The nearest case to the models
of the present paper, for which they give results, is for six flat extra dimensions. Thus
from the limits on M in their Table 1, we see that in 1998, the LEP 2 lower bound on
9 was around 107 GeV, and the Tevatron lower bound was around 125 GeV. And
the final lower bound on κ−
9 attainable at the Tevatron is expected to be around 166
GeV, and the final lower bound on κ−
9 attainable at the LHC is expected to be around
677 GeV. The relations between Mirabelli, Perelstein, and Peskin’s M , and Mp, the
Planck mass in D dimensions, as defined by Giddings and Thomas [275], and MD, the
Planck mass in D dimensions, as defined by Giudice, Rattazzi, and Wells [11], for the
case D = 11, and κ, are
M =Mp = 2
9MD = 2π
. (497)
Considering, now, the massless vector bosons in the effective theory in four dimen-
sions, we note that a smooth compact Einstein space of negative curvature cannot
have any continuous symmetries. For a vector field V A that generates a continuous
symmetry on a differentiable manifold M must satisfy the Killing vector equation
DAVB + DBVA = 0. Hence 0 = D
A (DAVB +DBVA). But from (5), on page 12, we
have DADBVA = DBD
AVA − RBDV D, and from the Killing vector equation, we have
DAVA = 0. And if M is an Einstein space of negative curvature, then RBD = αgBD,
where α > 0 is independent of position by the contracted Bianchi identity. Thus we
find DADAVB = αgBDV
D, hence V BDADAVB = αV
BgBDV
D. Thus if M is compact,
we find on integrating by parts that:
DAV B
(DAVB) = −α
ddxV BgBDV
D (498)
The left-hand side of this equation is ≥ 0, but for nonzero V A, the right-hand side
is < 0, so there can be no such nonzero V A. Thus since there is certainly no contin-
uous symmetry under translation in the radial direction of the thick pipe, the only
massless vector bosons in four dimensions, in the models considered in this paper, are
those which originate from the E8 Yang-Mills multiplets on the orbifold fixed-point
hyperplanes.
In standard compactifications of the weak coupling E8 heterotic superstring [97, 98],
the E8 containing the Standard Model [43, 44] is first broken to E6 by embedding the
spin connection in the gauge group [9, 72], and the E6 is then further broken by the
Hosotani mechanism [49, 50, 51]. However, in the models considered in the present
paper, the Standard Model is contained in the E8 on the inner surface of the thick
pipe, whereas if the compact six-manifold, M6, is a smooth compact quotient of CH3,
the spin connection is embedded in the E8 on the outer surface of the thick pipe, and
if M6 is a smooth compact quotient of H6, the spin connection is not embedded in
either of the two E8’s.
The fundamental group of M6 necessarily has no torsion in the sense of group
theory, or in other words, has no non-trivial finite subgroup, so if the vacuum contains
Hosotani configurations of the Yang-Mills fields, or in other words, topologically non-
trivial configurations of the Yang-Mills fields, with identically vanishing Yang-Mills
field strengths, they might have to be stabilized dynamically, by radiative corrections,
or partly dynamically and partly topologically, rather than purely topologically, as in
[9]. The dynamical Hosotani fields in the Cartan subalgebra of E8, analogous to the
Hosotani modes on a torus [49, 50, 51], would be proportional to harmonic 1-forms
on M6, which are associated with the non-torsion part of the first homology group
H1 (M6;Z), while Hosotani fields in the Cartan subalgebra of E8 that are associated
with the torsion part of H1 (M6;Z) would be partly topologically stabilized, and might
modify the potential for the dynamical Hosotani fields.
I shall assume that the first stage of breaking the E8 on the inner surface of the
thick pipe is by topologically non-trivial E8 vacuum gauge fields, localized on Hodge -
de Rham harmonic two-forms of M6, whose field strengths are topologically stabilized
in magnitude, and also partly in orientation within E8, by a form of Dirac quantization
condition, studied in subsection 5.3. When these Hodge - de Rham “monopoles” are all
in the Cartan subalgebra of E8, they break E8 either to SU (3)× (SU (2))3 × (U (1))3,
or to SU (3) × (SU (2))2 × (U (1))4, or to SU (3) × SU (2) × (U (1))5, and the U (1)’s,
other than U (1)Y , are also broken by a form of Higgs mechanism involving the CABy
components of the three-form gauge field, that was discussed by Witten [45], and by
Green, Schwarz, and West [372]. This arises, in the case of Hořava-Witten theory, from
the redefinition of GyUVW to include a term
δ (y − y1)ω(1)UVW , and an analogous
term involving δ (y − y2), in order to solve the modified Bianchi identity (42). Here
UVW is the Chern-Simons form constructed from the E8 gauge fields at y1:
UVW =tr
W −∂WA
V , A
+ cyclic permutations of U, V,W
(499)
This redefinition of GyUVW corresponds to the redefinition of the three-form field
strength of the two-form gauge field of N = 1 supergravity in ten dimensions, in
the Bergshoeff-de Roo-de Wit-van Nieuwenhuizen [109] and Chapline-Manton [110]
couplings of N = 1, d = 10 supergravity to Abelian gauge fields, and Yang-Mills fields,
respectively. ω
µAB contains a term 2tr
A(1)µ F
, and when F
AB has a vacuum ex-
pectation value in the Cartan subalgebra of E8, this leads, through the kinetic term
GIJKLG
IJKL of the three-form gauge field, to a mass term for the corresponding gauge
field in the Cartan subalgebra. However, when GyUVW is redefined as above, the re-
sulting ωUVWω
UVW term in the action is formally infinite, being proportional to δ (0),
so it would presumably be preferable to use Moss’s improved form of Hořava-Witten
theory [69, 70, 71], mentioned shortly after (47), on page 22, in which the δ (0) terms
are absent. It was noted by Witten, and by Green, Schwarz, and West, in the papers
cited above, that if the gauge field of a U (1) subgroup of E8 develops a vacuum expec-
tation value, but commutes with the gauge fields in the vacuum, it can be anomalous,
so consistency would require any such field that is anomalous to be massive also in
Moss’s form of the theory, so the δ (0) term would have to be replaced by a finite term,
rather than zero.
The Hodge - de Rham “monopoles” have non-vanishing Yang-Mills field strength,
and thus contribute to the vacuum energy on the inner surface of the thick pipe.
However, in the models considered in the present paper, the universe is stiffened by
effects largely determined by the region near the outer surface of the thick pipe, and in
particular, in the case studied in subsection 2.7, the universe is stiffened by the large
value of the integration constant G̃, defined in (352). Thus the presence of the Hodge
- de Rham monopoles, on the inner surface of the thick pipe, does not lead to a large
value of the effective cosmological constant in four dimensions.
When the Hodge - de Rham “monopoles” in the Cartan subalgebra break E8 directly
to SU (3) × SU (2) × (U (1))5, there is no need for any Hodge - de Rham monopoles
outside the Cartan subalgebra, but unification of the Yang-Mills coupling constants
then depends entirely on the accelerated unification mechanism studied by Dienes,
Dudas, and Gherghetta [128, 129], and by Arkani-Hamed, Cohen, and Georgi [373].
In this case, the Hodge - de Rham monopoles automatically satisfy the classical Yang-
Mills field equations.
When the Hodge - de Rham monopoles in the Cartan subalgebra break E8 to
SU (3)×(SU (2))2×(U (1))4, the (SU (2))2 must then be broken to the diagonal subgroup
SU (2)diag by monopoles outside the Cartan subalgebra, so that, at unification, the
Yang-Mills coupling constant of SU (2)diag is smaller than the Yang-Mills coupling
constant of SU (3), by a factor of 1√
, and the Yang-Mills coupling constants, as evolved
in the Standard Model, approximately unify at around 150 TeV, so there is still a need
for an accelerated unification effect, to achieve unification at around a TeV. The study
of the Dirac quantization condition, in subsection 5.3, only covers the case where all the
Hodge - de Rham monopoles are in the Cartan subalgebra, and I do not know whether
it is possible, by topological means, to prevent the Hodge - de Rham monopoles outside
the Cartan subalgebra, that break (SU (2))
to SU (2)diag, from “rotating”, or “sliding”,
back into the Cartan subalgebra. In the study of this case, I shall assume, without
proof, that this is possible.
And finally, when the Hodge - de Rham monopoles in the Cartan subalgebra break
E8 to SU (3)× (SU (2))3 × (U (1))3, the (SU (2))3 must also be broken to the diagonal
subgroup SU (2)diag by monopoles outside the Cartan subalgebra, so that, at unifi-
cation, the Yang-Mills coupling constant of SU (2)diag is smaller than the Yang-Mills
coupling constant of SU (3), by a factor of 1√
, and the SU (3) and SU (2)diag coupling
constants, as evolved in the Standard Model, now unify at around 413 GeV. However,
it is not possible to do this without breaking SU (2)diag×U (1)Y , and at the same time,
obtain an acceptable value of sin2 θW , which would have to be close to the value ≃ 0.23
observed at mZ , so this case appears to be excluded.
It is not possible to stabilize the absolute orientation of the Cartan subalgebra
within E8 topologically, and there will therefore, by Goldstone’s theorem [374], be
248 − 12 = 236 potentially massless Goldstone boson fields, corresponding to extra-
dimensional Lorentz components of the Yang-Mills fields, proportional to generators of
E8 outside the Standard Model SU (3))×SU (2)×U (1)Y , that can rotate different pos-
sible choices of the Standard Model SU (3))× SU (2)×U (1)Y into one another. These
modes, which are independent of position on M6, do not correspond to physical mass-
less Lorentz scalar multiplets, but rather become the longitudinal degrees of freedom
of the massive E8 gauge bosons outside the Standard Model SU (3))× SU (2)×U (1)Y
[375, 376, 377, 378].
I shall assume that M6 has first Betti number B1 > 0. There are then B1 linearly
independent harmonic 1-forms on M6, so that before the Dirac-quantized harmonic
2-form Hodge - de Rham monopoles in the E8 Cartan subalgebra are introduced, there
are at tree level B1 physical massless Lorentz scalar multiplets in the E8 fundamen-
tal/adjoint, one for each linearly independent harmonic 1-form. When E8 is broken
by the Hodge - de Rham monopoles, some of the resulting scalar multiplets have the
quantum numbers of the Standard Model Higgs field. The Hodge - de Rham monopoles
can also produce a potential for some or all of the scalar multiplets at tree level, which
can result in some of the scalars becoming tachyonic and developing vacuum expecta-
tion values, so that the Standard Model SU (3) × SU (2) × U (1)Y is broken as in the
ordinary Higgs effect [379, 380].
After the inclusion of radiative corrections, the potential is expected to depend on
all the scalar multiplets originating from harmonic 1-forms on M6, including any that
are not affected by the Hodge - de Rham monopoles, by the Coleman-Weinberg mech-
anism [381, 382], or equivalently, the Hosotani mechanism [49, 50, 51]. I shall assume
that this potential has a minimum in which a scalar multiplet with the quantum num-
bers of the Standard Model Higgs field has a vacuum expectation value, which after
integration over position on M6, produces masses for the Standard Model W± and Z
bosons, equivalent to the masses produced by the Standard Model Higgs boson with
a vacuum expectation value of 246 GeV ≃
2× 174 GeV, and breaks the electroweak
SU (2)×U (1)Y to U (1)e.m., as in the Standard Model. The original Coleman-Weinberg
mechanism resulted in a Higgs mass that was much smaller than the current exper-
imental lower bound of around 95 to 120 GeV, but more recent studies, taking into
account the large Yukawa coupling of the top quark, have found consistent solutions,
with a Higgs mass consistent with the current experimental constraints [383, 384, 385].
The vacuum expectation value of the scalar multiplet that serves as the Standard
Model Higgs field is proportional to a harmonic 1-form on M6, and is thus expected
to depend on position on M6. I shall assume that this enables the effective Yukawa
couplings of this scalar multiplet, identified as the Standard Model Higgs field, to
different pairs of chiral fermion zero modes to have different values, so as to realize
the fermion mass hierarchy, and the CKM [386, 387] and PMNS [388, 389] mixing
matrices, by a version of the Arkani-Hamed - Schmaltz mechanism [390]. I shall also
assume that all the other scalar multiplets that originate from harmonic 1-forms on
M6 are sufficiently massive at the minimum of the potential to be consistent with
experimental limits, even though they do not develop vacuum expectation values.
The Hodge - de Rham monopoles are required to satisfy Witten’s topological con-
straint [45], that was discussed in subsection 2.3.7. But since the vacuum field con-
figuration already satisfies this constraint in the absence of the Hodge - de Rham
monopoles, this means that the configuration of the Hodge - de Rham monopoles is
required to satisfy the requirement that for each closed four-dimensional submanifold
Q of the compact six-manifold M6, the integral
Q tr (F ∧ F ) is equal to zero. For a
given configuration F , of the E8 gauge fields on the inner surface of the thick pipe,
this integral only depends on the cohomology class of Q, and thus gives B4 constraints,
where B4 is the fourth Betti number of M6. But by Poincaré duality, B4 = B2, where
B2 is the second Betti number of M6. Hence there is one constraint per harmonic
two-form. However, the embedding of each harmonic two-form, in the Cartan subal-
gebra of E8, is determined by eight independent numbers, which, as I will show in
subsection 5.3, are constrained only to lie on a certain lattice in the Cartan subalgebra
of E8. Thus it seems likely that there will be non-trivial solutions of Witten’s topo-
logical constraint, even when the Hodge - de Rham monopoles are required to leave
SU (3)×(SU (2))n×(U (1))6−n unbroken, for the required value 3, 2, or 1, of n, and also
to be perpendicular to U (1)Y , so that the U (1)Y does not become massive by Witten’s
Higgs mechanism. However, when Witten’s topological constraint is imposed in addi-
tion to these requirements, there only remain 4− n degrees of freedom per monopole,
for the embedding in the E8 Cartan subalgebra, so the greatest flexibility is obtained
for n = 1.
The Hodge - de Rham monopoles result in the existence of chiral fermion zero
modes, for chiral fermions in various irreducible representations of the subgroup of E8
left unbroken by the monopoles and Witten’s Higgs mechanism involving the three-
form gauge field, and the number of chiral fermion zero modes, in each such irreducible
representation, is determined by the Atiyah-Singer index theorem [391]. Many of these
irreducible representations have the quantum numbers of a fermion representation in
the Standard Model, subject to the necessary accelerated unification of the Yang-Mills
coupling constants. And, as shown by Witten [45], and Green, Schwarz, and West
[372], Witten’s topological constraint ensures that there will be no gauge anomalies
involving only the gauge bosons left massless by the Hodge - de Rham monopoles and
Witten’s Higgs mechanism. Green, Schwarz, and West also state that the anomalies
involving the U (1) gauge bosons that commute with the vacuum Yang-Mills fields, but
become massive by Witten’s Higgs mechanism, due to having nonvanishing vacuum
expectation values themselves, are harmless.
For all the breakings of E8 considered in the present paper, there exists a U (1)
gauge boson Bµ that becomes massive by Witten’s Higgs mechanism, and one or more
irreducible representations with the quantum numbers of each left-handed fermion
representation in the Standard Model, such that the coupling of Bµ to each of those
fermion representations is a fixed multiple of the baryon number of that fermion repre-
sentation in the Standard Model. Sums of triangle diagrams with one or more external
Bµ’s are expected to be anomalous, but as explained by Witten [45], this does not
matter, due to the fact that Bµ has become massive by the Higgs mechanism involving
the CABy components of the three-form gauge field. Thus there might be a possibility
of stabilizing the proton in a manner similar to the Aranda-Carone mechanism [52],
although Aranda and Carone required the massive gauge boson, whose couplings to
the observed fermions are proportional to baryon number, to be non-anomalous.
In the case where the Hodge - de Rham monopoles in the Cartan subalgebra break
E8 directly to SU (3)×SU (2)× (U (1))5, and there are no Hodge - de Rham monopoles
outside the Cartan subalgebra, realizing the Standard Model requires:
1. finding a linear combination of the U (1)’s to serve as U (1)Y , such that there
exist SU (3)×SU (2) irreducible representations in the E8 fundamental, with the
correct SU (3)× SU (2) quantum numbers and U (1)Y charges to be identified as
the left-handed fermions of one or more generations, and the Higgs boson of the
Standard Model;
2. finding another linear combination of the U (1)’s to serve as U (1)B, such that
for each of the five types of SU (3)× SU (2) multiplet with non-vanishing U (1)Y
charge in the Standard Model, and also for the left-handed antineutrino, if these
are required, there exists at least one SU (3)× SU (2) irreducible representation
in the E8 fundamental, with those SU (3)× SU (2) quantum numbers and U (1)Y
charge, such that the U (1)B charge of that irreducible representation is a fixed
multiple of the baryon number of the corresponding fermion; and
3. finding, for each of the B2 linearly independent Hodge - de Rham harmonic two
forms of M6, where B2 is the second Betti number of M6, a point perpendicular
to U (1)Y , in the eight-dimensional lattice of points in the E8 Cartan subalgebra
that is allowed by the Dirac quantization condition, such that:
(a) Witten’s topological constraint is satisfied, for all B4 = B2 linearly indepen-
dent harmonic four-forms ofM6, or equivalently, for a set of B4 topologically
non-trivial closed four-dimensional surfaces in M6, linearly independent in
the sense of homology; and
(b) for each of the five or six types of SU (3)× SU (2) left-handed fermion mul-
tiplet in the Standard Model, depending on whether or not left-handed
antineutrinos are required:
i. every occurrence of that multiplet in the E8 fundamental, that has the
correct U (1)Y charge, and U (1)B charge equal to the correct multiple
of baryon number, has a net number of chiral fermion zero modes, as
given by the Atiyah-Singer index theorem, ≥ 0; and
ii. the sum, over all occurrences of that multiplet in the E8 fundamental,
that have the correct U (1)Y charge, and U (1)B charge equal to the
correct multiple of baryon number, of the net number of chiral fermion
zero modes, as given by the Atiyah-Singer index theorem, is equal to 3;
iii. every occurrence of that multiplet in the E8 fundamental, that either
has the wrong U (1)Y charge, or has U (1)B charge equal to the wrong
multiple of baryon number, has a net number of chiral fermion zero
modes, as given by the Atiyah-Singer index theorem, equal to 0;
(c) for each SU (3) × SU (2) multiplet in the E8 fundamental, that does not
correspond to a fermion multiplet in the Standard Model, or the complex
conjugate of a fermion multiplet in the Standard Model, the net number of
chiral fermion zero modes, as given by the Atiyah-Singer index theorem, is
equal to 0; and
(d) if there are sufficiently many left-handed antineutrinos, a Majorana mass
matrix, with one or more very light eigenstates by a generalized seesaw
mechanism, as discussed in subsection 5.7 below, is generated for them by
the Hodge - de Rham monopoles; and
(e) a potential is generated for all the “Higgs” bosons, by the Coleman-Weinberg
mechanism, that has a minimum at which all the “Higgs” bosons are mas-
sive, and the electrically neutral component of a “Higgs” boson, with the
quantum numbers of the Standard Model Higgs boson, has a vacuum expec-
tation value, possibly dependent on position on M6, whose value, averaged
over position on M6, produces masses for the Standard Model W± and Z
bosons, equivalent to the masses produced by the Standard Model Higgs
boson, with a vacuum expectation value of 246 GeV; and
(f) the mass matrices with entries given by the overlap integrals of pairs of
chiral fermion zero modes, with the vacuum expectation of the “Higgs”
boson, which may depend on position on M6, produce the observed mass
spectra of the quarks and the electrically charged leptons, and the CKM
mixing matrix of the quarks, by a version of the Arkani-Hamed - Schmalz
mechanism; and
(g) the masses of the Standard Model neutrinos, and the PMNS mixing matrix
of the Standard Model leptons, arise in some way.
In the present paper, I will present some solutions to the requirements 1. and 2.
above, both for the case when the Hodge - de Rhammonopoles in the Cartan subalgebra
of E8 break E8 directly to SU (3)×SU (2)× (U (1))5, and for the case when they break
E8 directly to SU (3)×(SU (2))2×(U (1))4. In the solutions where the Hodge - de Rham
monopoles in the Cartan subalgebra break E8 directly to SU (3)× (SU (2))2× (U (1))4,
there exist components of the E8 fundamental, outside the Cartan subalgebra, that
could break (SU (2))
to SU (2)diag, without breaking SU (3) × SU (2)diag × U (1)Y , if
they could be given topologically stabilized vacuum expectation values, as Hodge - de
Rham monopoles, but, as mentioned above, I do not know whether or not there is
any topological obstruction to prevent the orientation in E8, of such Hodge - de Rham
monopoles, from “rotating”, or “sliding”, back into the Cartan subalgebra.
The necessary first step for studying the requirements 3. (a) - (g) is to find explicit
examples of smooth compact quotients of CH3 or H6 that are spin manifolds. This is
unavoidable, because Witten’s topological constraint depends on the cohomology cup
product of the manifold [392, 393], that expresses the wedge product of pairs of har-
monic two-forms as linear combinations of harmonic four-forms, and this cohomology
cup product is a topological invariant of the manifold.
I shall now consider the lightest massive modes of the supergravity multiplet, in
the following subsection 5.1. The SU (9) basis for E8 is studied in subsection 5.2, on
page 235. The Dirac quantization condition on the field strengths of Hodge - de Rham
harmonic two-forms, in the Cartan subalgebra of E8, is studied in subsection 5.3, on
page 241. I show that there are no models with an acceptable value of sin2 θW , such
that the Hodge - de Rham monopoles, in the Cartan subalgebra of E8, break E8 to
SU (3)×(SU (2))3×(U (1))3, in subsection 5.4, on page 254. Models where the Hodge -
de Rham monopoles, in the Cartan subalgebra of E8, break E8 to SU (3)× (SU (2))2×
(U (1))
, are studied in subsection 5.5, on page 263, and models where they break E8
to SU (3)× SU (2)× (U (1))5, are studied in subsection 5.6, on page 270.
5.1 The lightest massive modes of the supergravity multiplet
From the point of view of the effective theory in four dimensions, supersymmetry is
broken explicitly in the models considered in the present paper, even though, from the
point of view of Hořava-Witten theory in eleven and ten dimensions, the supersymmetry
is broken spontaneously, by the compactification. Thus the gravitinos, four of which are
allowed, by the Hořava-Witten boundary conditions, to couple directly to the matter
on the inner surface of the thick pipe, and the associated spin-1
fermions, and also
the vectors and scalars which correspond, in four dimensions, to the three-form gauge
field, couple to ordinary matter with at most gravitational strength, and there is no
enhancement of the coupling of the gravitino to ordinary matter, as can happen in
models where N = 1 supersymmetry is broken spontaneously in four dimensions,
through the absorption of the goldstino by the gravitino [394, 395].
To study the Kaluza-Klein modes of the supergravity multiplet we have to expand
the quantum effective action to quadratic order in small fluctuations about the relevant
background solution, which is here one of the solutions found in subsections 2.5, 2.6,
and 2.7. For a first estimate I shall instead consider a massless scalar field Φ in the
bulk, which is intended to represent a small fluctuation of a component of any of the
supergravity fields, and retain only its classical action. Dropping also RΦ and H2Φ
terms, the equation for the small fluctuation Φ is then:
−GGIJ∂JΦ
= 0 (500)
Trying an ansatz Φ
xµ, xA, y
= ϕ (xµ)ψ
xA, y
, where coordinate indices µ, ν, σ, . . .
are tangent to the four observed space-time dimensions, and coordinate indices A,B,
C, . . . are tangent to M6, as in subsection 2.3, we find from (500) that:
(501)
b2ψ (xC , y)
hhAB∂Bψ
xC , y
a4b6ψ (xC , y)
a4b6∂yψ
xC , y
ϕ (xσ)
−ggµν∂νϕ (xσ)
The left-hand side of (501) is independent of xµ and the right-hand side is independent
of xA and y, hence each side must be a constant. The left-hand side is a positive
operator on a compact manifold so must be a non-negative constant m2 ≥ 0.
From the metric ansatz 94, on page 33, the metric Gµν at the inner surface of the
thick pipe, where we live, in the models considered here, is Gµν = A
dSgµν , where AdS
is the observed de Sitter radius 22, since by definition the de Sitter radius of gµν is
1. Thus in terms of the metric Gµν at the inner surface of the thick pipe, the wave
equation along the 4 extended dimensions, for a Kaluza-Klein mode ψ
xC , y
for which
each side of 501 is equal to m2, is:
−GGµν∂νϕ (xσ)
ϕ (xσ) = 0. (502)
For the solution found in subsection 2.5, starting on page 111, a and b are roughly
constant ∼ B over the main part of the classical region around y ∼ B, so there are
modes spread in this region for which −∂2yψ ∼ n
ψ, so that m ∼ n, for all integers
n > 0. Thus there are very light Kaluza-Klein modes of the bulk whose mass, as seen
at the inner surface of the thick pipe, is ∼ n
, for all integers n > 0. ψ
xC , y
suppressed in the region of the inner surface of the thick pipe for these modes, so the
situation is qualitatively similar to the situation considered by Randall and Sundrum
in [396], where the modifications to Einstein gravity in the 4 extended dimensions,
on the brane we live on, from modes of this form, were found to be negligibly small.
However the model considered in [396] did not include the ADD effect, so further study
would be needed to determine whether these very light Kaluza-Klein modes, localized
in the classical region of the bulk, prevent the solution found in subsection 2.5 from
being consistent with the precision Solar System tests of Einstein gravity [278, 279],
and with the sub-millimetre tests of Newton’s law [32].
For the solution found in subsection 2.6, starting on page 120, there are modes in the
second quantum region, adjacent to the outer surface of the thick pipe, that oscillate
sufficiently rapidly as y increases, that a4b6 is approximately constant over ∼ 10 or
more cycles, and wavepackets localized in this region can be formed from these modes.
For such a wavepacket localized at a ≃ acent and independent of position on M6, the
left-hand side of (501) is approximately −a
∂2yψ (y) ≃ m2, so a representative mode
is ψ (y) = cos my
acent
times a wavepacket profile. In this region a decreases exponentially
with increasing y
, with a coefficient ∼ 1 in the exponent, and b is a constant times
aτ̃ , where τ̃ is a constant of magnitude ∼ 1. Thus the requirement that a4b6 changes
over one wavelength by at most a factor close to 1 is that
(4+6τ̃)2πacent
κ2/9m
≪ 1. For
example m ∼ 103 would be adequate, for acent roughly at the outer boundary and
hence ∼ κ2/9. Thus from (502), there are very light Kaluza-Klein modes of the bulk
whose mass, as seen at the inner surface of the thick pipe, is ∼ 103 n
, for all integers
n > 0. ψ
xC , y
is again suppressed in the region of the inner surface of the thick pipe
for these modes, and further study would be needed to determine whether these modes
prevent the solution found in subsection 2.6 from being consistent with the precision
Solar System tests of Einstein gravity [278, 279], and with the sub-millimetre tests of
Newton’s law [32].
For the solution found in subsection 2.7, starting on page 137, where the outer
surface is stabilized in the classical region by fluxes, a and b are roughly constant, with
a ∼ 1022 metres, from (404), on page 155, and b ∼ B, over the main part of the classical
region around y ∼ B, so there are modes spread in this region for which −∂2yψ ∼ n
so that from (501), m ∼ n
× 1022 metres, for all integers n > 0. Thus from (502),
and (22), on page 15, the mass of these modes, as seen from the inner surface of the
thick pipe, is ∼ 10−4 n
, which from (399), on page 154, is ∼ n
10−8 metres
∼ 10n eV. The
wavefunctions of these modes are again suppressed in the region of the inner surface
of the thick pipe.
5.2 An SU (9) basis for E8
Throughout this section, I shall use an SU(9) basis for E8, as in [8]. On breaking E8 to
SU(9), the 248 of E8 splits to the 80, 84, and 84 of SU(9). Here the 80 is the adjoint
of SU(9), the 84 has three totally antisymmetrized SU(9) fundamental subscripts,
and the 84 has three totally antisymmetrized SU(9) antifundamental subscripts. The
fundamental representation generators (tα)ij̄ of SU (9) are normalized to satisfy [44]
tr (tαtβ) =
(503)
The generators of the required representations are as follows:
Antifundamental (Tα)īj = − (tα)jī
(504)
Adjoint (Tα)ij̄,k̄m = (tα)ik̄ δmj̄ − δik̄ (tα)mj̄
(505)
84 (Tα)ijk,m̄p̄q̄ =
((tα)im̄ δjp̄δkq̄ ± seventeen terms)
(506)
84 (Tα)̄ij̄k̄,mpq =
− (tα)mī δpj̄δqk̄ ± seventeen terms
(507)
where the additional terms in (506) and (507) antisymmetrize with respect to permu-
tations of (i, j, k), and with respect to permutations of (m, p, q). We can check directly
that these generators satisfy the same commutation relations as (tα)ij̄ , with the same
structure constants.
It is convenient to define:
δijk,r̄s̄t̄ ≡
(δir̄δjs̄δkt̄ + δis̄δjt̄δkr̄ + δit̄δjr̄δks̄ − δir̄δjt̄δks̄ − δis̄δjr̄δkt̄ − δit̄δjs̄δkr̄) (508)
which is the unit matrix in the space of matrices whose rows and columns are labelled
by antisymmetrized triples of indices, and projects expressions with three indices to
their antisymmetric part. Then we have:
(Tα)ijk,m̄p̄q̄ = δijk,r̄s̄t̄
(tα)rm̄ δsp̄δtq̄ + δrm̄ (tα)sp̄ δtq̄ + δrm̄δsp̄ (tα)tq̄
(tα)ir̄ δjs̄δkt̄ + δir̄ (tα)js̄ δkt̄ + δir̄δjs̄ (tα)kt̄
δrst,m̄p̄q̄ (509)
(Tα)̄ij̄k̄,mpq = −δrst,̄ij̄k̄
(tα)mr̄ δps̄δqt̄ + δmr̄ (tα)ps̄ δqt̄ + δmr̄δps̄ (tα)qt̄
(tα)rī δsj̄δtk̄ + δrī (tα)sj̄ δtk̄ + δrīδsj̄ (tα)tk̄
δmpq,r̄s̄t̄ (510)
We define the totally antisymmetric SU (9) structure constants fαβγ by [tα, tβ ] =
ifαβγtγ , noting, from (503), that the SU (9) generators tα, in the SU (9) fundamen-
tal representation, have been chosen to be hermitian. The generators of E8 are now
the 80 generators Tα of SU (9), together with 84 generators Trst, antisymmetric in rst,
whose label, rst, runs over the 84 of SU (9), and 84 generators Tr̄s̄t̄, antisymmetric in
r̄s̄t̄, whose label, r̄s̄t̄, runs over the 84 of SU (9). Indices A,B, C, . . . will run over all
248 generators of E8, as in the discussion of the SO (16) basis, in subsection 2.1. The
E8 structure constants will be written FABC , and defined such that [TA, TB] = iFABCTC ,
and in a similar way to the discussion of the SO (16) basis in subsection 2.1, I shall
use a summation convention such that each index in a multi-index, such as rst, gets
summed over its full range, without restrictions, and no compensating factor, such as
, is included. Thus when C, in iFABCTC, refers to r̄s̄t̄ on FABC , and to rst on TC ,
the contribution is iFAB,r̄s̄t̄Trst, with the normal summation convention, so that each
of the distinct generators Trst, 1 ≤ r < s < t ≤ 9, actually gets counted 6 times in
the sum. This is analogous to the convention used in the discussion of the SO (16)
basis in subsection 2.1, where the definition (34), of the orthogonal group structure
constants, means that the orthogonal group commutation relation (29) takes the form
[Jij , Jkl] = fij,kl,rsJrs, so that each of the distinct generators Jij , 1 ≤ i < j ≤ 16,
actually gets counted twice in the sum.
The structure constants FABC are totally antisymmetric under permutations of
A,B, C, and the non-vanishing matrix elements of the E8 generators are:
(Tα)βγ = ifβαγ = iFβαγ (511)
(Tα)ijk,m̄p̄q̄ = − (Tα)m̄p̄q̄,ijk = − (Tijk)α,m̄p̄q̄ = (Tijk)m̄p̄q̄,α =
= (Tm̄p̄q̄)α,ijk = − (Tm̄p̄q̄)ijk,α = iFijk,α,m̄p̄q̄ (512)
(Trst)ijk,mpq =
ǫrstijkmpq = iFijk,rst,mpq (513)
(Tr̄s̄t̄)̄ij̄k̄,m̄p̄q̄ =
ǫr̄s̄t̄̄ij̄k̄m̄p̄q̄ = iFīj̄k̄,r̄s̄t̄,m̄p̄q̄ (514)
The matrix representations of the generators are not antisymmetric in this basis, even
though the structure constants are totally antisymmetric, because it is necessary to
take the three types of index group in a different order for rows and columns, to ensure
that SU (9) anti-fundamental indices contract with SU (9) fundamental indices. The
matrix representations of the generators can be written as:
m̄p̄q̄ γ mpq
īj̄k̄
iFijk,α,m̄p̄q̄ 0 0
0 iFβαγ 0
0 0 iFīj̄k̄,α,mpq
(515)
Trst =
m̄p̄q̄ γ mpq
īj̄k̄
0 0 iFijk,rst,mpq
iFβ,rst,m̄p̄q̄ 0 0
0 iFīj̄k̄,rst,γ 0
(516)
Tr̄s̄t̄ =
m̄p̄q̄ γ mpq
īj̄k̄
0 iFijk,r̄s̄t̄,γ 0
0 0 iFβ,r̄s̄t̄,mpq
iFīj̄k̄,r̄s̄t̄,m̄p̄q̄ 0 0
(517)
To check the Jacobi identities, we first note that from the SU (9) commutation relation
for (Tα)ijk,m̄p̄q̄, we have:
Frst,α,m̄p̄q̄Fmpq,β,̄ij̄k̄ + Fīj̄k̄,α,mpqFm̄p̄q̄,rst,β + FβαγFγ,̄ij̄k̄,rst = 0 (518)
We next note that:
Fijk,α,ūv̄w̄Fuvw,rst,mpq + Fmpq,α,ūv̄w̄Fuvw,ijk,rst + Frst,α,ūv̄w̄Fuvw,mpq,ijk =
= − i
(tα)iū ǫrstujkmpq + (tα)jū ǫrstukimpq + (tα)kū ǫrstuijmpq
+ (tα)mū ǫrstijkupq + (tα)pū ǫrstijkuqm + (tα)qū ǫrstijkump
+ (tα)rū ǫustijkmpq + (tα)sū ǫutrijkmpq + (tα)tū ǫursijkmpq
(519)
The right-hand side is totally antisymmetric in rstijkmpq, for each value of α, and
is thus equal to an α-dependent multiple of ǫrstijkmpq. The α-dependent coefficient of
ǫrstijkmpq is found by contracting with ǫr̄s̄t̄̄ij̄k̄m̄p̄q̄, which gives zero, due to the traceless-
ness of tα.
Similarly, we find:
Fīj̄k̄,α,uvwFūv̄w̄,r̄s̄t̄,m̄p̄q̄ + Fm̄p̄q̄,α,uvwFūv̄w̄,̄ij̄k̄,r̄s̄t̄ + Fr̄s̄t̄,α,uvwFūv̄w̄,m̄p̄q̄,̄ij̄k̄ = 0 (520)
We next note that, due to the tracelessness and the normalization (503) of the SU (9)
generators, we have:
(tα)rī (tα)sj̄ =
δrj̄δs̄i −
δrīδsj̄
(521)
We now consider the expression:
Fuvw,̄ij̄k̄,αFα,m̄p̄q̄,rst + Frst,̄ij̄k̄,αFα,uvw,m̄p̄q̄ =
(tα)uī δvj̄δwk̄ ± seventeen terms
((tα)rm̄ δsp̄δtq̄ ± seventeen terms)
(tα)rī δsj̄δtk̄ ± seventeen terms
((tα)um̄ δvp̄δwq̄ ± seventeen terms) (522)
I will show that this is equal to:
δrīδsj̄δtk̄δum̄δvp̄δwq̄ ± 719 terms
ǭij̄k̄m̄p̄q̄x̄ȳz̄ǫrstuvwxyz = −Fm̄p̄q̄,̄ij̄k̄,x̄ȳz̄Fxyz,rst,uvw (523)
where the additional terms in the first line of (523) antisymmetrize with respect to
permutations of
i,j̄, k̄, m̄, p,q̄
. We first note that the terms in the first line of (523)
can be classified by the number n of elements of {u, v, w} that are joined by Kronecker
deltas to elements of {m̄, p,q̄}. We see that when we use (521) in the right-hand side
of (522), terms with n = 0 can only come from the second term in the right-hand side
of (521), used in the first term in the right-hand side of (522), terms with n = 1 can
only come from the first term in the right-hand side of (521), used in the first term in
the right-hand side of (522), terms with n = 2 can only come from the first term in
the right-hand side of (521), used in the second term in the right-hand side of (522),
and terms with n = 3 can only come from the second term in the right-hand side of
(521), used in the second term in the right-hand side of (522). Thus the first term in
the first line of (523), which has n = 3, comes from the second term in the right-hand
side of (521), used in the second term in the right-hand side of (522).
Considering, now, the coefficient of the first term in the first line of (523), we see
that it gets contributions from three terms in the first factor of the second term in the
right-hand side of (522), namely (tα)rī δsj̄δtk̄ + δrī (tα)sj̄ δtk̄ + δrīδsj̄ (tα)tk̄, times three
terms in the second factor of the second term in the right-hand side of (522), namely
(tα)um̄ δvp̄δwq̄+δum̄ (tα)vp̄ δwq̄+δum̄δvp̄ (tα)wq̄. Hence its coefficient is −
, as required.
Considering, next, the coefficient of a term with n = 2, namely δrm̄δsj̄δtk̄δuīδvp̄δwq̄,
in the first line of (523), we see that the locations of the tα’s are now fixed, and this
term only gets a contribution from the first term in the first factor of the second term
in the right-hand side of (522), times the first term in the second factor of the second
term in the right-hand side of (522). Hence its coefficient is − 1
= − 1
, as required.
And in a similar manner, we confirm the coefficient of a term with n = 1, namely
δum̄δvj̄δwk̄δrīδsp̄δtq̄, as
, and the coefficient of a term with n = 0, namely
δuīδvj̄δwk̄δrm̄δsp̄δtq̄, as − 172 . The coefficients of the remaining 716 terms in the first line
of (523), of which 62− 1 have n = 3, (34 × 22)− 1 have n = 2, (34 × 22)− 1 have n = 1
and 62 − 1 have n = 0, are then determined by the separate antisymmetries of the
left-hand side of (522) in (u, v, w), (r, s, t),
i,j̄, k̄
, and (m̄, p,q̄). And furthermore, all
182 + 182 terms in the right-hand side of (522) have now been accounted for. Thus we
find the final Jacobi identity:
Fuvw,̄ij̄k̄,αFα,m̄p̄q̄,rst + Frst,̄ij̄k̄,αFα,uvw,m̄p̄q̄ + Fm̄p̄q̄,̄ij̄k̄,x̄ȳz̄Fxyz,rst,uvw = 0 (524)
We next calculate Tr (TATB), where we recall, from just after (28), on page 19, that
we are using Hořava and Witten’s notation for traces in E8, so that for E8, “tr” denotes
of the trace in the adjoint representation, which is denoted by “Tr”. We also recall,
from above, our summation convention, that each index in a multi-index, such as rst,
gets summed over its full range, without restrictions, and no compensating factor, such
, is included.
We first note that, from (503), and the definition, [tα, tβ ] = ifαβγtγ , the SU (9)
structure constants fαβγ are given by fαβγ = −2itr ([tα, tβ] tγ). Hence we find:
fδαγfδβγ = −2tr ([tδ, tα] , [tδ, tβ]) = 9δαβ (525)
We next note that:
(Tα)ijk,m̄p̄q̄ (Tβ)mpq,̄ij̄k̄ =
= 3 (tα)rx̄ δsȳδtz̄
(tβ)xū δyv̄δzw̄ + δxū (tβ)yv̄ δzw̄ + δxūδyv̄ (tβ)zw̄
δuvw,r̄s̄t̄ =
δαβ (526)
Thus:
Tr (TαTβ) = (Tα)γδ (Tβ)δγ + (Tα)ijk,m̄p̄q̄ (Tβ)mpq,̄ij̄k̄ + (Tα)īj̄k̄,mpq (Tβ)m̄p̄q̄,ijk =
= 9δαβ +
δαβ +
δαβ = 30δαβ (527)
We next note that:
− Fijk,rst,mpqFm̄p̄q̄,ūv̄w̄,̄ij̄k̄ = −
ǫrstijkmpqǫūv̄w̄m̄p̄q̄īj̄k̄ = 10δrst,ūv̄w̄ (528)
And from (521), we find that:
−Fα,rst,m̄p̄q̄Fmpq,ūv̄w̄,α = 3δrst,f̄ ḡh̄ (tα)fx̄ δgȳδhz̄ (Tα)xyz,ūv̄w̄ =
− 1− 1
− 1− 1
δrst,ūv̄w̄ = 10δrst,ūv̄w̄ (529)
Thus:
Tr (TrstTūv̄w̄) = −Fijk,rst,mpqFm̄p̄q̄,ūv̄w̄,̄ij̄k̄ − Fα,rst,m̄p̄q̄Fmpq,ūv̄w̄,α − Fīj̄k̄,rst,βFβ,ūv̄w̄,ijk =
= (10 + 10 + 10) δrst,ūv̄w̄ = 30δrst,ūv̄w̄ (530)
And from the block matrix structure of the generators (515), (516), and (517), we see
Tr (TαTrst) = Tr (TαTr̄s̄t̄) = Tr (TrstTuvw) = Tr (Tr̄s̄t̄Tūv̄w̄) = 0 (531)
We note that we can choose a set of generators for the SU (9) Cartan subalgebra,
such that in the SU (9) fundamental, the generators of the Cartan subalgebra are diag-
onal matrices, and their nonzero matrix elements are equal to integers, times an overall
normalization factor that depends on the generator, and that there is an infinite variety
of such choices of the generators of the SU (9) Cartan subalgebra, consistent with (503).
And from (505), (506), (507), and (515), we see that for any such set of generators of
the SU (9) Cartan subalgebra, each generator of the E8 Cartan subalgebra, in the E8
fundamental / adjoint, will be a 248 × 248 diagonal matrix, whose nonzero matrix
elements are equal to integers, times an overall normalization factor that depends on
the generator.
The occurrence of the 84 and 84 of SU (9), in the fundamental of E8, is connected
to the presence of the three-form gauge field in d = 11 supergravity [38, 14], through
the embedding of SO (9), the little group of the d = 11 Poincare group, in E8, by
the subgroup chain SO (9) ⊂ SU (9) ⊂ E8. For, as reviewed in subsection 2.2 of
[397], half of the 32 supercharges of d = 11 supergravity vanish on the mass shell, and
the representation space of the 16 nonvanishing supercharges decomposes into the two
chiral spinor representations of SO (16), one of which corresponds to the bosonic states,
and the other to the fermionic states. The 16 nonvanishing spinor charges transform
according to a single spinor representation of the little group, SO (9), and the helicity
content of the bosonic and fermion states is determined by the branching of the two
different 128’s of SO (16), when SO (9) is embedded into SO (16) such that the spinor
of SO (9) becomes the vector of SO (16). This results in one of the 128’s of SO(16)
branching into the 44+84 of SO (9), corresponding to the graviton and the three-form
gauge field, while the other 128 of SO (16) becomes the 128 vector-spinor of SO (9),
corresponding to the gravitino, as can be checked by studying weight diagrams. On
the other hand, the adjoint of SO (16) branches into the antisymmetrized square of
the spinor of SO (9), which contains the 36 of SO (9), which is the adjoint, and the
84 of SO (9), which is the three-form. And under the embedding SO (9) ⊂ SU (9), the
adjoint of SU (9) branches to the adjoint and the 44 of SO (9), and the 84 and 84 of
SU (9) both become the 84 of SO (9). Thus the decomposition of the adjoint of E8
into irreducible representations of SO (9) is the same, when SO (9) is embedded into E8
according to the subgroup chains SO (9) ⊂ SU (9) ⊂ E8 and SO (9) ⊂ SO (16) ⊂ E8,
provided SO (9) is embedded into SO (16) in the manner that determines the helicity
content of the d = 11 supergravity states on the mass shell, and the spinor of SO (16),
in the fundamental of E8, is the one which branches to the 44+ 84 of SO (9).
5.3 Dirac quantization condition for E8 vacuum gauge fields
In this subsection, I will show that the field strengths of the Hodge - de Rham
monopoles are restricted in their possible magnitudes, and partly also in their pos-
sible orientations within E8, by a form of Dirac quantization condition. In particular,
if the configuration of the Yang-Mills fields is gauge equivalent to a configuration where
they lie everywhere within the Cartan subalgebra of E8, then for an arbitrary closed
smooth orientable two-dimensional surface S in the compact six-manifoldM6, the inte-
gral of the field strengths, over S, in a gauge where the Yang-Mills fields lie everywhere
within the Cartan subalgebra of E8, is a 248 × 248 diagonal matrix, that must be a
lattice point of a certain discrete lattice in the eight dimensional Cartan subalgebra of
We recall that for an arbitrary Yang-Mills gauge group, and for matter fields ψ
transforming in an arbitrary representation of the gauge group, with hermitian gener-
ators Tα satisfying [Tα, Tβ] = ifαβγTγ , with totally antisymmetric structure constants
fαβγ , the covariant derivative is Dµψ = (∂µ − igAµαTα)ψ, where Aµα are the Yang-
Mills fields and g is the coupling constant, and the Wilson line, or gauge covariant
path ordered phase factor, for a continuous path x (s), smin ≤ s ≤ smax, differentiable
except at a finite number of values of s, is:
W ({A},{x (s)})ij̄ =
(−ig)n
ds1 . . .
dsnθ (s1 − smin) θ (s2 − s1) . . . θ (sn − sn−1)×
×θ (smax − sn)
dxµ1 (s1)
. . .
dxµn (sn)
Aµ1α1 (x (s1)) . . . Aµnαn (x (sn)) (Tα1 . . . Tαn)ij̄ ,
(532)
where θ (s) is the step function, θ (s) = 1 for s ≥ 0, and θ (s) = 0 for s < 0. For paths
x1 (s), smin ≤ s ≤ smid, and x2 (s), smid ≤ s ≤ smax, such that x1 (smid) = x2 (smid),
W ({A} , {x (s)})ij̄ satisfies the product formula:
W ({A} , {x1 (s)})ik̄W ({A} , {x2 (s)})kj̄ =W ({A} , {x1 (s)} ∪ {x2 (s)})ij̄ (533)
where {x1 (s)} ∪ {x2 (s)} denotes the union of the two paths, which is a map with
domain smin ≤ s ≤ smax.
We now consider the transformations of Aµα, ψ, and W ({A} , {x (s)}), under fi-
nite gauge transformations, that might be topologically non-trivial, and might not be
connected to the identity. I shall assume that the gauge transformation parameters
Λα (x) are continuous and differentiable on each coordinate patch, and that the gauge
transformation acts on ψ by ψ (x) → U (x)ψ (x), where U (x) = eiΛ(x) = eiΛα(x)Tα .
Then the gauge-transformed Yang-Mills fields A′µα are required to satisfy Dµ (A
′)Uψ =
UDµ (A)ψ. Thus we require:
A′µαTα = AµαUTαU
(∂µU)U
† (534)
Using the Baker-Campbell-Hausdorff formula [398] eABe−A =B+[A,B]+ 1
[A, [A,B]]+
[A, [A, [A,B]]] + . . ., and also, for expanding eiΛ(x+δx)e−iΛ(x) to first order in δx, the
relation es(A+B)e−sA = 1 +
tABe−tAdt + O (B2), we find that (534) is satisfied for
an arbitrary representation with generators Tα, if:
A′µα = Aµβ
e−iΛ̆(x)
(∂µΛβ (x))
e−iΛ̆(x) − 1
−iΛ̆ (x)
(535)
where the matrix Λ̆βγ (x) is defined in terms of the generators
= −ifαβγ of
the adjoint representation, by Λ̆βγ (x) = Λα (x)
. The Wilson line (532) then
transforms as:
W ({A},{x (s)})ij̄ →W ({A
′},{x (s)})ij̄ = Uik̄ (x (smin))W ({A},{x (s)})kl̄ U
(x (smax))
(536)
Now on a topologically non-trivial manifold, such as the compact six-manifolds M6
considered in the present paper, the Yang-Mills fields can also be topologically non-
trivial. This means that Aµα (x) is not well-defined globally as a continuous and differ-
entiable function of the coordinates, which are themselves not defined globally. Instead
Aµα (x) is a continuous and differentiable function of the coordinates on each coordinate
patch, and where two patches i and j overlap, A(i)µα
is related to A
by both a general coordinate transformation, and a finite gauge transformation. This
is the case, for example, when the Yang-Mills fields are in the Cartan subalgebra of
the gauge group, and their field strengths are nonzero and proportional to Hodge - de
Rham harmonic two-forms.
The simplest example of this is a two-sphere centred on a Dirac magnetic monopole
[140] in the Wu-Yang gauge [399, 400]. The vector potential is tangential to the two-
sphere, and is well defined on two coordinate patches, one of which covers the northern
hemisphere, and a strip of the southern hemisphere along the equator, and the other
of which covers the southern hemisphere, and a strip of the northern hemisphere along
the equator. More generally, there will be three or more coordinate patches, and at any
point where three coordinate patches i, j, and k overlap, the gauge transformations
U i→j , U j→k, and Uk→i are required to satisfy U i→jU j→kUk→i = 1.
When a Wilson line crosses from a patch i to a patch j, we choose a point x on the
line in the overlap region between the two patches, at which to make the transition
from patch i to patch j, and the Wilson line is then defined to be the matrix product
of the segment of the line in patch i, and the segment of the line in patch j, as in
(533), but with the gauge transformation matrix U i→j (x) inserted between the two
segments. If we consider two different choices of the point x on the line in the overlap
region, at which to make the transition between the two patches, we find, from the
gauge transformation (536) of the segment of the Wilson line between the two different
choices of the transition point, that the Wilson line is independent of the choice of the
transition point.
Now if a Wilson line doubles back on itself like a hairpin, and exactly retraces its
path back to its starting point, then it is identically equal to the unit matrix, even
if the hairpin path crosses between several coordinate patches. And this is also true
for a hairpin path that has “branches”, that are themselves hairpins. Furthermore, by
the result just noted, this is also true if, for a segment of the hairpin path that lies in
the overlap region of two coordinate patches, we make the transition between the two
coordinate patches, at different points on the “outward” and “return” sections of the
hairpin path.
Let us now consider a configuration of the Yang-Mills fields that is gauge equivalent
to a configuration where the gauge fields are everywhere in the Cartan subalgebra of
the Lie algebra, and choose a gauge where the gauge fields are everywhere in the
Cartan subalgebra. Let us also assume that the manifold has non-vanishing second
Betti number, and that the gauge field configuration is topologically non-trivial, due
for example to including Hodge - de Rham harmonic two forms.
We now consider an arbitrary closed orientable two-dimensional surface in the man-
ifold, that is embedded in the manifold in a topologically non-trivial manner, in the
sense that it cannot be contracted to a point. Such surfaces exist due to the assumption
that the manifold has non-vanishing second Betti number. We do not know what the
intrinsic topology of the surface is, but it was shown by Seifert and Threlfall that the
most general closed orientable two-dimensional manifold is topologically equivalent to
a sphere with n handles, n ≥ 0. I shall consider a particular Wilson line that has the
form of a branched hairpin, and is thus equal to the identity matrix. However, the
hairpin branches will loop round and meet at their tips, in such a way that, due to
the assumption that the field configuration is Abelian, we can also express the Wilson
line as a diagonal matrix, such that each matrix element on the diagonal has the form
F , where
F denotes the integral over the closed two-dimensional surface, of the
two-form field strength of the corresponding diagonal matrix element of AµαTα. This
will be non-zero, if the field configuration includes a Hodge - de Rham harmonic two-
form, with non-zero coefficient in that matrix element, that has non-zero integral over
that surface. Thus g
F must be an integer multiple of 2π.
Considering, first, the case when the intrinsic topology of the two-dimensional sur-
face is an ordinary two-sphere, the intersections of the coordinate patches of the mani-
fold will define coordinate patches on the two-dimensional surface. Let us suppose, first,
that the coordinate patches on the two-dimensional surface are topologically equivalent
to the northern hemisphere, plus a strip of the southern hemisphere, and the southern
hemisphere, plus a strip of the northern hemisphere, as in the case of the Wu-Yang
gauge for the Dirac monopole. Then we choose a simple hairpin that starts at a point
on the equator, and wraps once round the equator, so that the point where the hairpin
doubles back on itself is the same as the point where it started. We choose the hairpin
to start on the northern hemisphere patch, and remain on the northern hemisphere
patch all the way around the equator to the point where it doubles back on itself, and
it makes the transition to the southern hemisphere patch at the point where it doubles
back on itself, and it remains on the southern hemisphere patch for the entire “return”
section of the hairpin, until it reaches the starting point, where it finally makes the
transition back to the northern hemisphere patch again. Then due to the Abelian na-
ture of the gauge field, each of the two transitions from one patch to the other simply
introduces a phase factor, and the two phase factors cancel one another because the two
transitions occurred at the same point. Furthermore, for an Abelian field configuration,
whose only non-vanishing matrix elements are on the diagonal, each non-vanishing ma-
trix element of the Wilson line has the form exp
dxµ(s)
Aµ (x (s))
, where Aµ
denotes the corresponding diagonal matrix element of AµαTα. We then uses Stokes’s
theorem to equate the line integral in the exponent, for the “outward” section of the
hairpin path, to the integral of F over the northern hemisphere, and the line integral
in the exponent, for the “return” section of the hairpin path, to the integral of F over
the southern hemisphere.
And if the coordinate patches on the two-dimensional surface, topologically equiva-
lent to a two-sphere, are not topologically equivalent to the northern hemisphere, plus
a strip of the southern hemisphere, and the southern hemisphere, plus a strip of the
northern hemisphere, we can introduce two new coordinate patches in the manifold,
whose intersections with the two-dimensional surface do have this form, and choose
suitable gauges on these two coordinate patches, such that we can use the intersections
of these two coordinate patches with the two-dimensional surface, as the coordinate
patches on the two-dimensional surface, and then use the argument as above.
Considering, now, the case where the intrinsic topology of the two-dimensional
surface is a sphere with n handles, n ≥ 1, it will be sufficient to show that we can
always find a suitable branched hairpin, that divides the surface into suitable sectors,
so that we can use the same arguments as above. We note, first, that we can always
cut a sphere with n handles, n ≥ 1, in such as way as to transform it into a polygon
with 4n sides, such that opposite sides are identified. Figure 3 (i) shows a way of doing
this for n = 3, that extends directly to all n ≥ 1. In this diagram, paired circles AA,
BB, and CC are identified by reflection in the vertical midline of the diagram, to form
✭✭✭❳❳❳
✟✟✟✘✘✘✘
❅❤❤❤❤✭✭✭✭✟
❤❤❤❤❤
✭✭✭❤❤❤
❳❳❳❳❍❍❍❍
✑✑✭✭✭✭✭✭❛❛
S S′Q
✏✏✏✏✏✏✏PP
✟✟✟✟✟✟✟✟
✻ ❄ ❏
✟✟✟✙✟
PPPPPPP
✏✏✏✏✏✏✏
▲▲ ✁✁
✏✏✏✏✮
❈❈❈❈❲
❅❅❘❅❅❅
PPPPPq
✏✏✶✏✏
✔✔ ❆❆✔✔
✔✔ ✔✔
❍❍❍❍❍❍❍ �� ✦✦
❜❜ ✦✦
✟✟✟✟✟✟✟✟
Figure 3: (i) A sphere with three handles, cut so as to transform it into a twelve-sided
polygon with opposite sides identified. (ii) A multi-hairpin Wilson line for a sphere
with three handles. Opposite sides of the polygon, for example P and P′, are identified.
handles, and the remaining lines are the cuts. Figure 3 (ii) shows a branched hairpin
dividing the sphere with three handles into twelve triangular regions, which we can
assume correspond to the main parts of the coordinate patches on the two-dimensional
surface in this case. We make the transitions between the coordinate patches, such
that the three sections of the Wilson line directly surrounding each triangle, are on the
coordinate patch corresponding to that triangle.
The individual branches of the hairpin all branch out of the Wilson line at a single
point, which is the central point of Figure 3 (i), and corresponds to all twelve vertices
of the polygon in Figure 3 (ii). Six of the 6 + 12− 1 = 17 hairpins that branch out of
this point loop round and meet this point again at their tips. These are the hairpins
PP ′, QQ′, RR′, SS ′, TT ′, and UU ′, along the edges of the polygon in Figure 3 (ii).
The Wilson line starts and ends at a different point, corresponding to the centre of
the polygon in Figure 3 (ii), which could be any other point of the sphere with three
handles shown in Figure 3 (i), and the remaining 12− 1 = 11 hairpins, which are the
hairpins running from vertices b to l of the polygon in Figure 3 (ii), to the centre of
that polygon, also loop round to meet that point at their tips. These hairpins reach
that point in Figure 3 (i), by passing along the handles, as necessary. For example, if
the Wilson line starts and ends at a point somewhere in the external region of Figure
3 (i), the hairpin that runs from vertex b of the polygon in Figure 3 (ii), to the centre
of that polygon, reaches that point from b in Figure 3 (i), by first passing along handle
A, then along handle B, and finally along handle C.
If we label a hairpin that runs from a vertex of the polygon in Figure 3 (ii) to
the centre of that polygon, by the letter of the corresponding vertex, then after the
initial section from the centre of the polygon to vertex a, the Wilson line runs along
the hairpins in the sequence PP ′, h, c, RR′, j, e, TT ′, l, g, b, QQ′, i, d, SS ′, k, f, UU ′, then
finally along the final section from vertex a back to the centre of the polygon. We see
that each transition, from one coordinate patch to another, that occurs across a side
of the polygon in Figure 3 (ii), is matched by a reverse transition through the same
point, so that all the phase factors associated with these transitions cancel out. While
for the transitions at the centre of the polygon in Figure 3 (ii), we see that, since the
Wilson line must end with a transition back to the coordinate patch it started on, we
have transitions corresponding to diagonal matrices U1→2, U2→3, . . . , U12→1, all at the
same point, where the patches are labelled 1 to 12 anticlockwise around the polygon,
and the product of all these is equal to 1. Furthermore, each of the twelve triangular
regions is circled anticlockwise by the Wilson line sections around its edge, which are
the sections of the Wilson line on the coordinate patch corresponding to that triangle,
so we can use Stokes’s theorem for each triangle.
Considering, now, how this works for general n ≥ 1, we draw the corresponding
4n-sided polygon with an opposite pair of its vertices pointing east and west. We draw
a T , consisting of the initial and final sections of the Wilson line, and two half hairpins,
with the centre of its top at the easternmost vertex, as in Figure 3 (ii). And for each of
the remaining 2n−1 sides of the upper half of the polygon, we draw an L, consisting of
one and a half hairpins, with the foot of the L pointing anticlockwise as in Figure 3 (ii).
And for each of the remaining 2n−1 sides of the lower half of the polygon, we draw an
L, consisting of one and a half hairpins, with the foot of the L pointing clockwise, as in
Figure 3 (ii). And finally we draw an I, consisting of a single hairpin, with its foot at
the westernmost vertex, as in Figure 3 (ii). We draw an arrow pointing anticlockwise
on every Wilson line section running along an edge of the polygon, as in Figure 3 (ii),
and add arrows to the Wilson line sections directly joined to these sections, consistent
with these arrows, so that every triangular section is circled anticlockwise by the three
Wilson line sections around its edge.
The cancellation of the phase factors associated with the transitions between co-
ordinate patches, and the use of Stokes’s theorem, will now work exactly as for the
n = 3 case, so it remains to check that, starting at the start of the Wilson line, we
pass along each Wilson line section exactly once, and in the correct direction. To check
this, we number the Wilson line sections running along the perimeter of the top half
of the polygon 0, 1, . . . , (2n− 1) in sequence anticlockwise, starting at the easternmost
section, which is half the top of the T , and labelled P in Figure 3 (ii). And we number
the Wilson line sections running along the perimeter of the lower half of the poly-
gon 0′, 1′, . . . , (2n− 1)′ in sequence anticlockwise, starting at the westernmost section,
which is labelled P ′ in Figure 3 (ii). Thus the L’s in the top half of the polygon are
numbered 1, 2, . . . (2n− 1), and the L’s in the lower half of the polygon are numbered
0′, 1′, . . . , (2n− 2)′.
We observe that, due to the directions of the arrows on the Wilson line sections,
each pair of opposite L’s of the form mm′, 1 ≤ m ≤ (2n− 2), is traversed in the
sequence: first m, then m′. Furthermore, the upper half of the top of the T , labelled P
in Figure 3 (ii), and 0 in the general numbering scheme, is traversed immediately after
the initial section of the Wilson line, and immediately before the L labelled 0′, which is
labelled P ′ in Figure 3 (ii), and the lower half of the top of the T , labelled U ′ in Figure
3 (ii), and (2n− 1)′ in the general numbering scheme, is traversed immediately after
the L labelled (2n− 1), which is labelled U in Figure 3 (ii), and immediately before
the final section of the Wilson line. Furthermore, the hairpin based at the westernmost
vertex of the polygon, labelled g in Figure 3 (ii), is traversed immediately after the
L labelled (2n− 2)′, which is labelled T ′ in Figure 3 (ii), and immediately before the
L labelled 1, which is labelled Q in Figure 3 (ii). And finally, for 0 ≤ m ≤ (2n− 3),
L number m′, in the lower half of the polygon, is immediately followed by L number
(m+ 2), in the upper half of the polygon.
Thus the 4n Wilson line sections running along the perimeter of the polygon, and
the Wilson line sections directly connected to them in the diagram, and the hairpin
based at the westernmost vertex of the polygon, which together comprise the 4n +
1 pieces of Wilson line that are directly connected in the diagram, are traversed in
the sequence: 0, 0′, 2, 2′, 4, 4′, . . . , (2n− 2) , (2n− 2)′, then the hairpin based at the
westernmost vertex of the polygon, then 1, 1′, 3, 3′, 5, 5′, . . . , (2n− 1) , (2n− 1)′.
If there is just one coordinate patch, as is natural when a compact hyperbolic
manifold is specified by giving a Dirichlet domain for it in uncompactified hyperbolic
space, together with the face-pairing maps for the Dirichlet domain, a simpler tree of
hairpins can be obtained from the one shown in Figure 3 (ii), by moving the start and
end point to just inside the 12-sided polygon at a, and shrinking the eleven hairpins
that meet at the centre of the polygon, back to the perimeter of the polygon, so that
all that remains are the hairpin halves around the perimeter of the polygon, which are
traversed in the sequence PP ′RR′TT ′QQ′SS ′UU ′.
Thus we have shown that if the configuration of the Yang-Mills fields lies entirely
within the Cartan subalgebra of the gauge group, then for an arbitrary representation
of the gauge group, with generators Tα, such that matter fields exist that transform
under that representation of the gauge group, and for each matrix element on the
leading diagonal of that representation of the gauge group, and for an arbitrary closed
orientable two-dimensional surface embedded smoothly in the manifold, the integral
F must be an integer multiple of 2π, where
F denotes the integral over the closed
two-dimensional surface, of the two-form field strength of the corresponding diagonal
matrix element of AµαTα. And we noted that this integral will be non-zero, if the field
configuration includes a Hodge - de Rham harmonic two-form, with non-zero coefficient
in that matrix element, that has non-zero integral over that surface.
Now for the fundamental / adjoint representation of E8, in the SU (9) basis used
in this section, each of the eight generators of the Cartan subalgebra of E8, which are
the eight generators of the Cartan subalgebra of SU (9), in a reducible representation
of SU (9) that comprises the 80, 84, and 84 of SU (9), is such that its nonzero ma-
trix elements are integer multiples of an overall normalization factor, specific to that
generator. Let us now consider Aµα such that α denotes a fixed one of the eight gen-
erators of the Cartan subalgebra of E8. Let B2 denote the second Betti number of the
manifold, which by assumption is > 0. Then there are B2 linearly independent Hodge
- de Rham harmonic two-forms, and there are also just B2 non-contractible closed
two-dimensional surfaces in the manifold, that are linearly independent in the sense of
homology. Thus we can choose a basis of B2 non-contractible closed two-dimensional
surfaces in the manifold, such that for an arbitrary closed two-form F , or in other
words, for an arbitrary two-form F that satisfies the Bianchi identity dF = 0, or in
components, ∂[µFνσ] = 0, and an arbitrary closed two-dimensional surface in the man-
ifold, the integral
F , of F over the surface, is equal to a linear combination of the
corresponding integrals for the B2 surfaces in the basis.
Thus if we consider one particular matrix element in the diagonal of Tµα, for the
particular α in the Cartan subalgebra we are considering, and restrict Aµα to be a
linear combination of the one-form vector potentials of the B2 harmonic two-forms,
so that the Yang-Mills field equations will automatically be satisfied, for this Abelian
field configuration, there are just B2 linearly independent quantization conditions, to
be satisfied by B2 independent coefficients. And if we now extend the consideration
to all 248 matrix elements on the leading diagonal of Tµα, but still for the fixed value
of α in the Cartan subalgebra, we see that, because the ratios of the matrix elements
are fixed rational numbers, there will be a finite integer p, such that if Aµα satisfies
the quantization condition for one particular matrix element on the diagonal, such that
that matrix element of Tµα is nonzero, then pAµα will satisfy the quantization condition
for all the nonzero matrix elements of the diagonal matrix Tµα.
Thus, still considering Aµα for just one fixed value of α in the Cartan subalgebra,
the quantization condition can be satisfied by an infinite number of non-trivial field
configurations, and for field configurations that satisfy the Yang-Mills field equations,
and are thus a linear combination of the B2 Hodge - de Rham harmonic two-forms,
the allowed values of the coefficients of the B2 Hodge - de Rham harmonic two-forms
will lie on a discrete B2-dimensional lattice, because B2 linearly independent linear
combinations of the coefficients have quantized values, so that after a suitable change
of basis, each coefficient would be quantized independently. And when we choose such
a basis for the B2 Hodge - de Rham harmonic two-forms, so that we can consider each
Hodge - de Rham harmonic two-form in the basis independently, the allowed values of
the Aµα, associated with any one Hodge - de Rham harmonic two form, will be integer
multiples of a basic “monopole”.
Let us now choose such a basis for the B2 Hodge - de Rham harmonic two-forms,
and consider one Hodge - de Rham harmonic two-form in the basis, so that the allowed
values of Aµα will be integer multiples of a basic “monopole”. We now allow Aµα to be
nonzero for all the eight values of α in the Cartan subalgebra. Then the solutions of the
quantization condition will include, in particular, a discrete eight-dimensional lattice,
in the Cartan subalgebra of E8, whose lattice points correspond to Yang-Mills fields of
the form of the sum over the Cartan subalgebra of qαAµαTα, where the eight qα are the
integers that define the lattice point, and Aµα (x) is the Hodge - de Rham harmonic two-
form under consideration, times a normalization factor, dependent on α, that makes it
into the correspnonding basic “monopole”, for the element α of the Cartan subalgebra.
There may now be additional solutions of the quantization conditions, such that some
or all of the qα are non-integer rational numbers, but the number of such additional
solutions, in each unit cell of the lattice defined by integer qα, will be finite, since 248
linear combinations of the eight qα, not necessarily all distinct, are required to satisfy
quantization conditions, which are, however, mutually consistent, and among these
248 linear combinations, there are eight that are linearly independent. Furthermore,
given a point in the eight-dimensional space of the qα, that satisfies all the quantization
conditions, and such that not all eight of the qα are integers, other non-integer solutions
of the quantization conditions can be obtained by adding arbitrary integers to the qα.
Thus the solutions of the quantization conditions form an infinite discrete lattice in
the space of the qα, which is, however, not necessarily hypercubic.
Thus, for each separate Hodge - de Rham harmonic two-form, in a basis in which we
can apply the quantization conditions to each separate Hodge - de Rham harmonic two-
form independently, we can have Abelian configurations of the E8 Yang-Mills fields,
that solve the classical Yang-Mills field equations, and, within the Cartan subalgebra,
are topologically stabilized, and whose field strengths have the spatial dependence of
the Hodge - de Rham harmonic two form, and an embedding within E8, that lies on
any lattice point of an infinite eight-dimensional lattice in the eight-dimensional Cartan
subalgebra of E8. Thus, provided the different lattice points are not connected to one
another by orbits within E8 that go outside the Cartan subalgebra, we can break E8 to
a wide variety of subgroups, in a topologically stabilized manner, by introducing such
Hodge - de Rham harmonic two-forms in the vacuum, embedded in E8 on suitable
lattice points of this infinite eight-dimensional lattice in the eight-dimensional Cartan
subalgebra of E8.
Furthermore, for breakings to smaller subgroups of E8, such as the subgroups
SU (3)×(SU (2))3×(U (1))3, SU (3)×(SU (2))2×(U (1))4, and SU (3)×SU (2)×(U (1))5,
considered in this paper, there is a multi-dimensional space of embeddings in the Car-
tan subalgebra of E8, that achieve the required breaking. Thus we can choose a differ-
ent embedding, consistent with the required breaking, for each independent Hodge - de
Rham harmonic two-form, subject to the requirement of satisfying Witten’s topological
constraint [45], as discussed in subsection 2.3.7, and thus seek to satisfy the conditions
3. (a) to (g), in the list above. Specifically, for breaking to SU (3)×(SU (2))3×(U (1))3,
the space of embeddings that achieve the required breaking is three-dimensional, while
for breaking to SU (3)× (SU (2))2× (U (1))4, it is four-dimensional, and for breaking to
SU (3)×SU (2)×(U (1))5, it is five-dimensional. However, in each case, we also need to
ensure that the embeddings of all the monopoles are perpendicular to U (1)Y , so that
U (1)Y is not broken by Witten’s Higgs mechanism involving the components CABy
of the three-form gauge field [45], which reduces the dimensionalities of the spaces of
available embeddings to two, three, and four, respectively. And if we want to make
the unwanted U (1)’s massive by Witten’s Higgs mechanism, rather than by monopoles
outside the Cartan subalgebra, we also have to ensure that the embeddings of at least
some of the monopoles are not perpendicular to the unwanted U (1)’s.
We can ensure that we really do get the expected multi-dimensional lattice of em-
beddings within the E8 Cartan subalgebra, consistent with the required breaking, by
choosing a basis for the Cartan subalgebra such that a certain subset of the gen-
erators automatically preserve the required subgroup. For example, the subgroup
SU (3) × (SU (2))3 × (U (1))3 is preserved by an arbitrary element of the Cartan sub-
algebra of E8, which in the basis used in the present section, is also the Cartan sub-
algebra of SU (9), whose diagonal matrix elements, in the SU (9) fundamental, are
(σ1, σ1, σ1, σ2, σ2, σ3, σ3, σ4, σ4), with 3σ1 + 2σ2 + 2σ3 + 2σ4 = 0. Of course, for cer-
tain values of the σi, a larger subgroup is preserved. For example, (σ1, σ2, σ3, σ4) =
(2,−1,−1,−1) preserves E7, (0, 0, 1,−1) preserves SO (14), (4, 4,−5,−5) preserves
SU (5) × SU (4), (2, 2, 2,−7) preserves SU (7) × SU (2), (2, 2,−1,−4) preserves E6 ×
SU (2), (4,−5, 1,−2) preserves SO (10) × SU (3), and (0, 1,−2, 1) preserves SU (3) ×
SO (10). However, for most of the points of the lattice, which in this case is three-
dimensional, the required breaking is obtained.
This partial topological stabilization no longer applies for continuous variations of
the gauge field configuration that are allowed to go outside the Cartan subalgebra. For
a generic path that has the same start and end point as the tree of hairpins, and is
homotopic to the tree of hairpins, the logarithm F ≡ lnW , of the Wilson phase factor
W for the path, can generically be defined by continuity under continuous variations
of the path and of the gauge field configuration. For an assumed small variation δ of
lnW , we have:
W ′ = ei(F+δ) = eiF +
dseiF siδeiF (1−s) + . . . =
= eiF + i
eiF̆ − 1
iF + . . . , (537)
where F̆βγ = Fα(T̆α)βγ = −iFαfαβγ is in the adjoint representation, and the Baker-
Campbell-Hausdorff formula has been used as in the derivation of (535). Thus δ
could fail to be determined by W ′ if F̆ has one or more eigenvalues equal to non-zero
multiples of 2π, and this must inevitably happen for variations of a path that transform
it into the tree of hairpins, for we can transform the tree of hairpins continuously to
the trivial path by continuously retracting the hairpins. Thus lnW is undefined for the
tree of hairpins, for general variations of the gauge field configuration that go outside
the Cartan subalgebra.
The above discussion has involved only the components of the gauge field tangential
to a particular closed smooth orientable two-dimensional surface S embedded in M6,
and for this restricted system, the question of the existence of any possible absolute
topological stabilization of a non-trivial configuration of the gauge field reduces to the
corresponding question for S. From subsection 4.1 of [401], if the gauge group G is
connected and simply connected, then a G bundle over a two-dimensional surface is
trivial. From [402], the compact Lie group E8 is simply connected and appears also to
be connected. Furthermore, it has trivial centre, so it is not a covering group of any
other connected Lie group [403]. Thus the Dirac-quantized harmonic 2-formmonopoles
considered in this subsection are apparently not absolutely stabilized topologically,
although they might be separated by potential energy barriers from other solutions of
the classical Yang-Mills equations, including pure gauge configurations.
5.4 Nonexistence of models where the Abelian Hodge - de
Rham monopoles break E8 to SU (3)× (SU (2))3 × (U (1))3
In models of TeV-scale gravity based on Hořava-Witten theory, such as those considered
in the present paper, the gauge couplings have to unify at around a TeV, rather than at
1016 GeV. One way this could work is if the running of the coupling constants somehow
accelerates, so that the couplings run to their conventional unification values at the
TeV scale, rather than at 1016 GeV. This possibility was studied by Dienes, Dudas,
and Gherghetta [128, 129], and by Arkani-Hamed, Cohen, and Georgi [373].
An alternative possibility, considered in [8], is to embed SU(3)× SU(2)×U(1) into
E8 in an unusual way, so that the values of the coupling constants, at unification, are
equal to their observed values, as evolved conventionally to the TeV scale. Usually the
coupling constant of a simple non-Abelian subgroup of a Grand Unification group, at
unification, is equal to the coupling constant of the Grand Unification group, irrespec-
tive of how the subgroup is embedded in the Grand Unification group. An exception
occurs [6, 373] if the initial breaking of the Grand Unification group produces N copies
of the simple subgroup, and the N copies of the simple subgroup then break into their
“diagonal” subgroup. In this case, after the second stage of the breaking, the coupling
constant of the “diagonal” subgroup is equal to 1√
times the coupling constant of
the Grand Unification group. Effectively, the gauge field, in each of the N copies of
the simple non-Abelian subgroup, becomes equal to 1√
times the “diagonal” gauge
field, plus massive vector terms that can be ignored at low energies. The sum of the
N copies of the Yang-Mills action, of the simple non-Abelian subgroup, then becomes
equal to the Yang-Mills action of the “diagonal” subgroup, whose coupling constant is
times the coupling constant of the Grand Unification group.
Looking at the observed values of the reciprocals of the SU(3)× SU(2)×U(1) fine
structure constants, at MZ , normalized so as to meet at unification in SU(5) Grand
Unification, [404], (Mohapatra [405], page 22):
α−13 (MZ) = 8.47± .22
α−12 (MZ) = 29.61± .05
α−11 (MZ) = 58.97± .05
(538)
we see that they are quite close to being in the ratios 1, 3, 6.
If we evolve them in the Standard Model, [44], then α−13 and α
2 reach an exact
ratio of 1, 3, at 413 GeV, at which point α−13 is equal to 10.12. At this point, α
First generation LH states
Multiplet Y
SU(3)× SU(2)
content
SU(5) coefficient
required
coefficient
uR uG uB
dR dG dB
(3,2) 1√
ūR ūG ūB
(3̄, 1) −4√
d̄R d̄G d̄B
(3̄, 1) 2√
 −1 (1,2) −3√
2 (1,1) 6√
0 (1,1) absent 0
Table 6: Weak hypercharge, SU(3)× SU(2) assignments, coefficient of the coupling to
the U(1)Y vector boson in SU(5), and the required coefficient of the coupling to the
U(1)Y vector boson, for the left-handed fermions of the first generation.
equal to 58.00, which is 4% off being 6 times α−13 , and sin
2 θW ≃ 0.239.
Thus it is natural to consider the breaking of E8 to SU(3) × (SU(2))3 × U(1)Y ,
followed by the breaking of (SU(2))3 to SU(2)diag, and seek an embedding of U(1)Y
that gives the correct hypercharges at unification. I have summarized the required
left-handed fermions of the first generation, together with their hypercharges, Y , [44],
the coefficients of their U(1)Y couplings in SU(5) Grand Unification, and the required
coefficients of their U(1)Y couplings, in Table 6. Here I have assumed that α
3 and
α−11 are in the ratio 1, 6, at unification, but it would be useful to study models that
achieve this within a few percent, since the correct form of running to unification is not
yet known. Since the running of the coupling constants is always by small amounts,
the additional states in these models, not yet observed experimentally, will not alter
the unification mass, or the value of the SU(3) coupling constant at unification, which
is equal to the E8 coupling constant at unification, by a large amount. Thus this class
of models generically predicts that the unification mass is about a TeV, and the E8
fine structure constant, at unification, is about 1
The breaking of E8 to SU(3)× (SU(2))3 × U(1)Y can be studied, following [8], by
analyzing the breaking of SU(9) to SU(3)× (SU(2))3 × U(1)Y . It is convenient to use
block matrix notation. Each SU(9) fundamental index is replaced by a pair of indexes,
an upper-case letter and a lower-case letter. The upper-case letter runs from 1 to 4,
and indicates which subgroup in the sequence SU(3)×SU(2)×SU(2)×SU(2) the block
belongs to. Thus B = 1 denotes the SU(3), B = 2 denotes the first SU(2), B = 3
denotes the second SU(2), and B = 4 denotes the third SU(2). The lower-case index
is a fundamental index for the subgroup identified by the upper-case index it belongs
to. It is important to note that the range of a lower-case index depends on the value
of the upper-case index it belongs to, so we have to keep track of which lower-case
indexes belong to which upper-case indexes. Each SU(9) antifundamental index is
treated in the same way, except that the lower-case index is now an antifundamental
index for the appropriate subgroup. The summation convention is applied to both
upper-case letters and lower-case letters that derive from an SU(9) fundamental or
antifundamental index, but we have to remember that lower-case indexes are to be
summed over first, because their ranges of summation depend on the values of the
upper-case indexes they belong to. Each SU(9) adjoint representation index, which in
the notation above, is a lower-case Greek letter, is replaced by a pair of indexes, an
upper-case letter and a lower-case letter, where the upper-case letter runs from 1 to 5,
and identifies which subgroup in the sequence SU(3)×SU(2)×SU(2)×SU(2)×U(1)Y
a generator belongs to, and the lower-case letter runs over all the generators of the
subgroup identified by the upper-case letter it belongs to. When an upper-case adjoint
representation index takes the value 5, the associated lower-case index takes a single
value, 1. The summation convention is applied to a lower-case letter that derives from
an SU(9) adjoint representation index, but not to an upper-case letter that derives
from an SU(9) adjoint representation index.
We can now list all the blocks in the 80, the 84, and the 84, and display their
SU(3)× SU(2) content. This is displayed in Table 7 for the 80, and in Table 8 for the
84, with all the lower-case indexes suppressed.
The SU(9) generators, in the SU(3) × SU(2) × SU(2) × SU(2) × U(1)Y subgroup,
may be taken as follows, in the block matrix notation.
BiC̄j̄
= δABδAC̄ (tAa)ij̄ (1 ≤ A ≤ 4) (539)
BiC̄j̄
σ1δ1Bδ1C̄δij̄ + σ2δ2Bδ2C̄δij̄ + σ3δ3Bδ3C̄δij̄ + σ4δ4Bδ4C̄δij̄
σAδABδAC̄δij̄
(540)
Here (tAa)ij̄ denotes the fundamental representation generator number a, of non-
States in the 80
Blocks
Number of
distinct
blocks
SU(3)× SU(2)
content
Number of
states
coefficient
of coupling
to U(1)
ψ11̄ 1 (8,1) 8 0
ψ22̄ 1 (1,3) 3 0
ψ33̄ 1 (1,3) 3 0
ψ44̄ 1 (1,3) 3 0
ψdiag
ψdiag
ψdiag
applicable
(1, 1) + (1, 1)+
+(1, 1)
ψ12̄ 1 (3,2) 6
σ1−σ2
ψ13̄ 1 (3,2) 6
σ1−σ3
ψ14̄ 1 (3,2) 6
σ1−σ4
ψ21̄ 1 (3̄, 2) 6
−σ1+σ2
ψ31̄ 1 (3̄, 2) 6
−σ1+σ3
ψ41̄ 1 (3̄, 2) 6
−σ1+σ4
ψ23̄ 1 (1, 3) + (1, 1) 4
σ2−σ3
ψ24̄ 1 (1, 3) + (1, 1) 4
σ2−σ4
ψ34̄ 1 (1, 3) + (1, 1) 4
σ3−σ4
ψ32̄ 1 (1, 3) + (1, 1) 4
−σ2+σ3
ψ42̄ 1 (1, 3) + (1, 1) 4
−σ2+σ4
ψ43̄ 1 (1, 3) + (1, 1) 4
−σ3+σ4
Table 7: The states in the 80, organized by their SU(3) × SU(2) × SU(2) × SU(2)
assignments, showing their SU(3) × SU(2)diag content, and the coefficients of their
couplings to a U(1) gauge field, parametrized as in equation (540).
States in the 84
Blocks
Number of
distinct
blocks
SU(3)× SU(2)
content
Number of
states
coefficient
of coupling
to U(1)
ψ1̄1̄1̄ 1 (1,1) 1
ψ2̄1̄1̄ ψ1̄2̄1̄ ψ1̄1̄2̄ 1 (3, 2) 6
−2σ1−σ2
ψ3̄1̄1̄ ψ1̄3̄1̄ ψ1̄1̄3̄ 1 (3, 2) 6
−2σ1−σ3
ψ4̄1̄1̄ ψ1̄4̄1̄ ψ1̄1̄4̄ 1 (3, 2) 6
−2σ1−σ4
ψ2̄2̄1̄ ψ2̄1̄2̄ ψ1̄2̄2̄ 1 (3̄, 1) 3
−σ1−2σ2
ψ3̄3̄1̄ ψ3̄1̄3̄ ψ1̄3̄3̄ 1 (3̄, 1) 3
−σ1−2σ3
ψ4̄4̄1̄ ψ4̄1̄4̄ ψ1̄4̄4̄ 1 (3̄, 1) 3
−σ1−2σ4
ψ1̄2̄3̄ ψ2̄1̄3̄ ψ2̄3̄1̄
ψ1̄3̄2̄ ψ3̄1̄2̄ ψ3̄2̄1̄
1 (3̄, 3) + (3̄, 1) 12 −σ1−σ2−σ3
ψ1̄2̄4̄ ψ2̄1̄4̄ ψ2̄4̄1̄
ψ1̄4̄2̄ ψ4̄1̄2̄ ψ4̄2̄1̄
1 (3̄, 3) + (3̄, 1) 12 −σ1−σ2−σ4
ψ1̄3̄4̄ ψ3̄1̄4̄ ψ3̄4̄1̄
ψ1̄4̄3̄ ψ4̄1̄3̄ ψ4̄3̄1̄
1 (3̄, 3) + (3̄, 1) 12 −σ1−σ3−σ4
ψ2̄2̄2̄ ψ3̄3̄3̄ ψ4̄4̄4̄ these three blocks are empty
ψ2̄2̄3̄ ψ2̄3̄2̄ ψ3̄2̄2̄ 1 (1,2) 2
−2σ2−σ3
ψ2̄2̄4̄ ψ2̄4̄2̄ ψ4̄2̄2̄ 1 (1,2) 2
−2σ2−σ4
ψ3̄3̄2̄ ψ3̄2̄3̄ ψ2̄3̄3̄ 1 (1,2) 2
−σ2−2σ3
ψ3̄3̄4̄ ψ3̄4̄3̄ ψ4̄3̄3̄ 1 (1,2) 2
−2σ3−σ4
ψ4̄4̄2̄ ψ4̄2̄4̄ ψ2̄4̄4̄ 1 (1,2) 2
−σ2−2σ4
ψ4̄4̄3̄ ψ4̄3̄4̄ ψ3̄4̄4̄ 1 (1,2) 2
−σ3−2σ4
ψ2̄3̄4̄ ψ3̄2̄4̄ ψ3̄4̄2̄
ψ4̄3̄2̄ ψ4̄2̄3̄ ψ2̄4̄3̄
(1, 4) + (1, 2)+
+(1, 2)
8 −σ2−σ3−σ4
Table 8: The states in the 84, organized by their SU(3) × SU(2) × SU(2) × SU(2)
assignments, showing their SU(3) × SU(2)diag content, and the coefficients of their
couplings to a U(1) gauge field, parametrized as in equation (540).
Abelian subgroup number A, in the list above. Thus for A = 1, the subgroup is SU(3),
a runs from 1 to 8, and i and j each run from 1 to 3, while for A = 2, 3, or 4, the
subgroup is SU(2), a runs from 1 to 3, and i and j each run from 1 to 2.
σ1, σ2, σ3, and σ4 are real numbers parametrizing the embedding of the U(1)Y
subgroup in SU(9), and thus in E8, and θ is a normalization factor.
In using this notation, we have to take sensible precautions, such as grouping within
brackets, to keep track of which lower-case indexes belong to which upper-case indexes.
In equation (540), it would be wrong to “factor out” the δij̄ , because it represents a
3 by 3 matrix in one term, and a 2 by 2 matrix in the other three terms.
The tracelessness of
BiC̄j̄
implies:
0 = 3σ1 + 2 (σ2 + σ3 + σ4) (541)
and the normalization condition (503) implies:
θ2 = 6σ21 + 4
σ22 + σ
3 + σ
(542)
As an example, I consider the states in the left-handed 84. The covariant derivative
is [44]
Dµ = ∂µ − igAµαTα (543)
so, for unbroken SU(9), and with (+,+,+, 1) metric, and {γµ, γν} = 2gµν , the massless
Dirac action in this case is [44]:
ψ̄γµDµψ = ψ̄γµ (∂µ − igAµαTα)ψ =
= ψ̄ijkγ
δmīδpj̄δqk̄ ± five terms
−igAµα
− (tα)mī δpj̄δqk̄ ± seventeen terms
ψm̄p̄q̄ =
= ψ̄ijkγ
µ∂µψīj̄k̄ + 3igAµαψ̄ijkγ
µ (tα)mī ψm̄j̄k̄ (544)
where I used (507), the antisymmetry of ψ̄ijk and ψm̄p̄q̄ in their indexes, and the re-
labelling of dummy indexes. ψ̄ijk are the right-handed 84 states, and ψmpq are the
left-handed 84 states.
Breaking SU(9) to SU(3)× (SU(2))3×U(1)Y , and using the block matrix notation,
this becomes:
ψ̄BiCjDkγ
µ∂µψB̄īC̄j̄D̄k̄ + 3ig
AµAaψ̄BiCjDkγ
EmB̄ī
ψĒm̄C̄j̄D̄k̄ =
ψ̄BiCjDkγ
µ∂µψB̄īC̄j̄D̄k̄ + 3ig
AµAaψ̄AiCjDkγ
µ (tAa)mī ψĀm̄C̄j̄D̄k̄
+3igAµY
σAψ̄AiCjDkγ
µψĀīC̄j̄D̄k̄
(545)
where I used (539) and (540). We can now extract the covariant derivative Dirac action
terms for the various entries in Table 8, and thus determine the coefficients of their
couplings to AµY . For example, a block in ψB̄īC̄j̄D̄k̄, where two upper-case indexes take
the value 1, and the remaining upper-case index takes the value 2, 3, or 4, is a candidate
to be a (3,2) quark multiplet. The sum of all terms in (545), where two upper-case
indexes take the value 1, and the remaining upper-case index takes the value 2, is:
3ψ̄1i1j2kγ
µ∂µψ1̄ī1̄j̄2̄k̄+6igAµ1aψ̄1i1j2kγ
µ(t1a)mī ψ1̄m̄1̄j̄2̄k̄+3igAµ2aψ̄2i1j1kγ
µ(t2a)mī ψ2̄m̄1̄j̄1̄k̄
+6igAµY
σ1ψ̄1i1j2kγ
µψ1̄ī1̄j̄2̄k̄ + 3igAµY
σ2ψ̄2i1j1kγ
µψ2̄ī1̄j̄1̄k̄
(546)
Now ψ1̄ī1̄j̄2̄k̄ is antisymmetric under swapping the two SU(3) antifundamental in-
dexes i and j, so that we may write:
ψ1̄ī1̄j̄2̄k̄ = εīj̄m̄φmk̄ (547)
and analogously:
ψ̄1i1j2k = εijmφ̄m̄k (548)
We can then use relations such as
ψ̄1i1j2kγ
µψ1̄ī1̄j̄2̄k̄ = εijpφ̄p̄kγ
µεīj̄m̄φmk̄ = 2δpm̄φ̄p̄kγ
µφmk̄ = 2φ̄m̄kγ
µφmk̄ (549)
ψ̄1i1j2kγ
µ (t1a)mī ψ1̄m̄1̄j̄2̄k̄ = εijpφ̄p̄kγ
µ (t1a)mī εm̄j̄q̄φqk̄ =
= (δim̄δpq̄ − δiq̄δpm̄) φ̄p̄kγµ (t1a)mī φqk̄ = φ̄p̄kγ
µ (t1a)īi φpk̄ − φ̄m̄kγ
µ (t1a)mī φik̄ =
= −φ̄m̄kγµ (t1a)mī φik̄ (550)
to express (546) as:
φ̄ījγ
µ∂µφij̄ − igAµ1aφ̄ījγµ (t1a)ik̄ φkj̄ + igAµ2aφ̄ījγ
µ(t2a)mj̄ φim̄
−igAµY
(−2σ1 − σ2) φ̄ījγµφij̄
(551)
Thus we see that the index i of φij̄ is an SU(3) fundamental index. The SU(2)
antifundamental is equivalent to the fundamental, the relation being given by matrix
multiplication by εjk, and we could, if we wished, make a further transformation to
replace the SU(2) antifundamental index j̄ of φij̄, by an index that is manifestly in
the SU(2) fundamental. When (SU(2))3 is broken to SU(2)diag, the Aµ2a, in the third
term in (551), will be replaced, at low energy, by 1√
Bµa, where Bµa is the gauge field of
SU(2)diag. The overall factor of 6 can be absorbed into the normalizations of φij̄ and φ̄īj ,
so from the fourth term in (551), we can read off what the coefficient of gAµY φ̄ījγ
µφij̄
would be, if the φ̄īijiγ
µ∂µφij̄ term had standard normalization, and thus complete the
entries in the second row of Table 8.
The entries in the third column of Tables 7 and 8 can be completed by similar
methods. The entries in the fifth column of Table 8 can be completed by a simple
mnemonic: for each upper-case index, of the untransformed ψB̄īC̄j̄D̄k̄, that takes the
value A, 1 ≤ A ≤ 4, include a term −1
σA. For Table 7, the mnemonic is that when
the index B of ψBiC̄j̄ takes the value A, 1 ≤ A ≤ 4, so that i is in the fundamental
of non-Abelian subgroup number A, include a term 1
σA, and when the index C̄ of
ψBiC̄j̄ takes the value A, 1 ≤ A ≤ 4, so that j is in the antifundamental of non-Abelian
subgroup number A, include a term −1
Indeed, suppose we extract all terms from (545) such that ψB̄īC̄j̄D̄k̄ has nA upper-
case indexes with the value A, 1 ≤ A ≤ 4, so that n1 + n2 + n3 + n4 = 3. We get
n1!n2!n3!n4!
contributions from the ψ̄BiCjDkγ
µ∂µψB̄īC̄j̄D̄k̄ term. The number of times we
get σA, from the third term in (545), is
ñ1!ñ2!ñ3!ñ4!
, where ñB = nB if B 6= A, and
ñA = nA − 1. But this is equal to 2!nAn1!n2!n3!n4! . The factor
n1!n2!n3!n4!
combines with the
explicit factor of 3, in the third term in (545), to produce the same overall factor of
n1!n2!n3!n4!
as found for the first term, so the coefficient of the contributions from the
third term, if the contributions from the first term had standard normalization, would
be −1
nAσA. The mnemonic for Table 7 can be justified in a similar manner.
We know that we have to find couplings of the observed fermions, to the U(1)Y
gauge field, that are smaller than those found in the SU(5) model [404, 405], by an
overall factor that is within a few percent of 1√
, so it is useful to apply the same
techniques to calculate the corresponding coefficients in the SU(5) model. In this case,
the relations (503) and (541) completely determine the U(1) generator, up to sign, and
we find the entries in the fourth column of Table 6. The entries in the fifth column
have been filled in, assuming the overall factor is exactly 1√
If we now choose (σ1, σ2, σ3, σ4) = (4, 3,−3,−6), so that θ2 = 312, we can identify
ψ12̄ as a q, the (3̄, 1) state in ψ1̄2̄3̄ as a ū, ψ3̄3̄1̄ as a d̄, ψ2̄2̄3̄ as an l, and the (1, 1) state in
ψ23̄ as an e
+. We note that another U (1), defined by (σB1, σB2, σB3, σB4) = (2, 0, 0,−3),
couples to these states in proportion to their baryon number. Furthermore, ψ3̄3̄2̄, which
has Y = 1, can be identified as the Standard Model Higgs field.
To determine the value of sin2 θW at unification, for this choice of the σi, let us
denote the Higgs field, ψ3̄3̄2̄, by φi. Then by the methods above, we find that at low
energies, its covariant derivative, times i, becomes:
i∂µφi + g
(σa)ik φk + g
AµY φi, (552)
where Bµa is the gauge field of SU (2)diag. While from equation (117) on page 33 of
Rosner’s review [44], the standard covariant derivative, times i, on the Standard Model
Higgs field is:
i∂µφi + gRosnerBµa
(σa)ik φk + g
Rosner
AµY φi (553)
Thus since the Standard Model Higgs field has Y = 1, we see that:
gRosner = g
, g′Rosner = g
(554)
Now by definition, tan θW =
Rosner
gRosner
. Thus we find that, for this choice of the σi,
sin2 θW =
≃ 0.257 at unification. This is the closest I have found to the required
value of sin2 θW ≃ 0.239, when the Hodge - de Rham harmonic two-forms, in the
Cartan subalgebra, break E8 to SU (3)× (SU (2))3 × (U (1))3.
However, even this value of sin2 θW cannot actually be realized. For to break
(SU (2))
to SU (2)diag, without breaking SU (3)× SU (2)diag × U (1)Y , we need to find
states in the E8 fundamental / adjoint, that transform nontrivially under (SU (2))
but are singlets of SU (3) × SU (2)diag, and have Y = 0. The only states which
transform nontrivially under (SU (2))
, but are singlets of SU (3)× SU (2)diag, are the
(1, 1) states in ψ23̄, ψ24̄, ψ34̄, and their complex conjugates. Looking at the U (1)
couplings of these states, in Table 7, we see that none of them have Y = 0, for
(σ1, σ2, σ3, σ4) = (4, 3,−3,−6). Furthermore, to ensure that SU (2)diag is the diagonal
subgroup of all three SU (2)’s, and not just two of them, we need at least two of ψ23̄,
ψ24̄, and ψ34̄, to have Y = 0. That means we require σ1 = σ2 = σ3, which means it
is impossible to have ū and d̄ states with different values of Y . Thus we cannot real-
ize the Standard Model, if the Hodge - de Rham harmonic two-forms, in the Cartan
subalgebra of E8, break E8 to SU (3)× (SU (2))3 × (U (1))3.
5.5 Models where the Abelian Hodge - de Rham monopoles
break E8 to SU (3)× (SU (2))2 × (U (1))4
In this subsection, I shall consider some models where the Hodge - de Rham harmonic
two-forms break E8 to SU (3)×(SU (2))2×(U (1))4. We will find that there are states of
E8 that transform nontrivially under (SU (2))
, and can break (SU (2))
to SU (2)diag,
without breaking SU (3) × SU (2)diag × U (1)Y , for a reasonable value of the U (1)Y
coupling constant at unification. The SU (2)diag coupling constant is now
at uni-
fication, so the SU (3) and SU (2)diag coupling constants, as evolved in the Standard
Model, now meet at around 145 TeV, although this could presumably be reduced to
around a TeV by the accelerated unification mechanism [128, 129, 373]. We will find
two distinct types of solution for U (1)Y , both of which give sin
2 θW =
= 0.300 at
unification, roughly halfway between the observed value ≃ 0.23 at mZ , and the value
= 0.375 found in conventional grand unification [404]. The observed value of sin2 θW
evolves to ≃ 0.270 at around 145 TeV, in the Standard Model.
An element of the E8 Cartan subalgebra, and hence of the SU (9) Cartan subalgebra,
that can have a vacuum expectation value without breaking this subgroup of E8, is, in
the SU (9) fundamental, a diagonal matrix, with diagonal matrix elements
(σ1, σ1, σ1, σ2, σ2, σ3, σ3, σ4, σ5) (555)
such that:
3σ1 + 2σ2 + 2σ3 + σ4 + σ5 = 0 (556)
The normalization condition is now:
θ2 = 6σ21 + 4
σ22 + σ
σ24 + σ
(557)
The states in the 80 are shown in Table 9, and the states in the 84 are shown in Table
10. To break (SU (2))
to SU (2)diag, without breaking SU (3)× SU (2)diag ×U (1)Y , we
need an SU (3) singlet state that transforms non-trivially under (SU (2))
, but contains
a singlet of SU (2)diag, to have Y = 0, so that the singlet of SU (2)diag can have a
non-vanishing vacuum expectation value, without breaking U (1)Y . Thus at least one
of ψ23̄, ψ2̄3̄4̄, and ψ2̄3̄5̄ is required to have vanishing U (1)Y charge, so at least one of
(σ2 − σ3), (−σ2 − σ3 − σ4), and (−σ2 − σ3 − σ5) is required to be zero.
There are nine SU (3) × SU (2)diag singlets, plus their complex conjugates, whose
U (1) charges do not vanish identically. However, only three of the U (1) charges of
these nine SU (3)× SU (2)diag singlets are linearly independent, so it is not possible to
raise the masses of more than two of the three unwanted U (1)’s as much as required,
without breaking SU (3)× SU (2)diag × U (1)Y , and without relying on Witten’s Higgs
mechanism.
I did a computer search to determine whether the number of distinct choices of
U (1)Y , such that there is at least one set of q, u, d, l, and e states with the correct
relative Y values, and such that two (1, 1) states of SU (3)×SU (2)diag, with independent
U (1)Y charges, have Y = 0, is finite or infinite. Specifically, I generated all sets of
integer-valued (σ1, σ2, σ3, σ4), in order of increasing |σ1|+ |σ2|+ |σ3|+ |σ4|, up to |σ1|+
|σ2|+ |σ3|+ |σ4| = 300, with σ5 fixed by (556), and tested for the existence of at least
one set of (3, 2), (3, 1), (3, 1), (1, 2), and (1, 1) states of SU (3)×SU (2)diag, with U (1)Y
charges in the ratios 1, 4,−2,±3,±6, respectively. The result was that, excluding
(σ1, σ2, σ3, σ4) with greatest common divisor > 1, and solutions related to solutions
already found, by multiplying by −1, or by swapping σ2 and σ3, or by swapping σ4
and σ5, thirteen distinct solutions were found with |σ1| + |σ2| + |σ3| + |σ4| ≤ 10, and
no new solutions were found with 11 ≤ |σ1| + |σ2| + |σ3| + |σ4| ≤ 300. Thus it looks
likely that the thirteen distinct solutions, found with |σ1|+ |σ2|+ |σ3|+ |σ4| ≤ 10, are
the only distinct solutions.
All thirteen solutions were found to satisfy the requirement that at least one of
(σ2 − σ3), (−σ2 − σ3 − σ4), and (−σ2 − σ3 − σ5) is zero, so that (SU (2))2 can be broken
to SU (2)diag, without breaking SU (3)× SU (2)diag × U (1)Y . Furthermore, all thirteen
solutions were found to admit a choice of a set of q, u, d, l, and e states with the
correct relative Y values, such that there exists a U (1)B, defined by a different set
of σi, whose couplings to that set of q, u, d, l, and e states are proportional to their
baryon number, so that there is a chance of stabilizing the proton by a version of the
Aranda-Carone mechanism. For a given set of q, u, d, l, and e states, the requirement
for such a U (1)B to exist is four homogeneous linear equations for (σ1, σ2, σ3, σ4), and
is thus equivalent to the vanishing of the determinant of the matrix of the coefficients
of these equations.
To try to find out if any of the thirteen solutions might be physically equivalent to
Block SU (3)× SU (2)diag Coupling 3Y1 3Y2 3Y3 3B3
ψ11̄ (8, 1) 8 0 0 0 0 0
ψ22̄ (1, 3) 3 0 0 0 0 0
ψ33̄ (1, 3) 3 0 0 0 0 0
ψS (1, 1) 1 0 0 0 0 0
ψT (1, 1) 1 0 0 0 0 0
ψX (1, 1) 1 0 0 0 0 0
ψY (1, 1) 1 0 0 0 0 0
ψ12̄ (3, 2) 6
σ1−σ2
1 q 1 q 1 q 1
ψ13̄ (3, 2) 6
σ1−σ3
1 q 1 q −5 1
ψ14̄ (3, 1) 3
σ1−σ4
1 4 u −2 d 1
ψ15̄ (3, 1) 3
σ1−σ5
−5 10 −8 4
ψ21̄ (3̄, 2) 6
−σ1+σ2
−1 q̄ −1 q̄ −1 q̄ −1
ψ31̄ (3̄, 2) 6
−σ1+σ3
−1 q̄ −1 q̄ 5 −1
ψ41̄ (3̄, 1) 3
−σ1+σ4
−1 −4 ū 2 d̄ −1
ψ51̄ (3̄, 1) 3
−σ1+σ5
5 −10 8 −4
ψ23̄ (1, 3) + (1, 1) 4
σ2−σ3
0 ν̄ 0 ν̄ −6 e− 0
ψ24̄ (1, 2) 2
σ2−σ4
0 3 l̄ −3 l 0
ψ25̄ (1, 2) 2
σ2−σ5
−6 9 −9 3
ψ34̄ (1, 2) 2
σ3−σ4
0 3 l̄ 3 l̄ 0
ψ35̄ (1, 2) 2
σ3−σ5
−6 9 −3 l 3
ψ45̄ (1, 1) 1
σ4−σ5
−6 e− 6 e+ −6 e− 3
ψ32̄ (1, 3) + (1, 1) 4
−σ2+σ3
0 ν̄ 0 ν̄ 6 e+ 0
ψ42̄ (1, 2) 2
−σ2+σ4
0 −3 l 3 l̄ 0
ψ52̄ (1, 2) 2
−σ2+σ5
6 −9 9 −3
ψ43̄ (1, 2) 2
−σ3+σ4
0 −3 l −3 l 0
ψ53̄ (1, 2) 2
−σ3+σ5
6 −9 3 l̄ −3
ψ54̄ (1, 1) 1
−σ4+σ5
6 e+ −6 e− 6 e+ −3
Table 9: The states in the 80 for the SU (3)× (SU (2))2 × (U (1))4 case.
Block SU (3)× SU (2)diag Coupling 3Y1 3Y2 3Y3 3B3
ψ1̄1̄1̄ (1, 1) 1
0 ν̄ −6 e− 6 e+ −3
ψ1̄1̄2̄ (3, 2) 6
−2σ1−σ2
1 q −5 7 −2
ψ1̄1̄3̄ (3, 2) 6
−2σ1−σ3
1 q −5 1 q −2
ψ1̄1̄4̄ (3, 1) 3
−2σ1−σ4
1 −2 d 4 u −2
ψ1̄1̄5̄ (3, 1) 3
−2σ1−σ5
−5 4 u −2 d 1
ψ1̄2̄2̄ (3̄, 1) 3
−σ1−2σ2
2 d̄ −4 ū 8 −1
ψ1̄3̄3̄ (3̄, 1) 3
−σ1−2σ3
2 d̄ −4 ū −4 ū −1
ψ1̄2̄3̄ (3̄, 3) + (3̄, 1) 12
−σ1−σ2−σ3
2 d̄ −4 ū 2 d̄ −1
ψ1̄2̄4̄ (3̄, 2) 6
−σ1−σ2−σ4
2 −1 q̄ 5 −1
ψ1̄2̄5̄ (3̄, 2) 6
−σ1−σ2−σ5
−4 5 −1 q̄ 2
ψ1̄3̄4̄ (3̄, 2) 6
−σ1−σ3−σ4
2 −1 q̄ −1 q̄ −1
ψ1̄3̄5̄ (3̄, 2) 6
−σ1−σ3−σ5
−4 5 −7 2
ψ1̄4̄5̄ (3̄, 1) 3
−σ1−σ4−σ5
−4 ū 8 −4 ū 2
ψ2̄2̄3̄ (1, 2) 2
−2σ2−σ3
3 l̄ −3 l 3 l̄ 0
ψ2̄3̄3̄ (1, 2) 2
−σ2−2σ3
3 l̄ −3 l −3 l 0
ψ2̄2̄4̄ (1, 1) 1
−2σ2−σ4
3 0 ν̄ 6 e+ 0
ψ2̄2̄5̄ (1, 1) 1
−2σ2−σ5
−3 6 e+ 0 ν̄ 3
ψ3̄3̄4̄ (1, 1) 1
−2σ3−σ4
3 0 ν̄ −6 e− 0
ψ3̄3̄5̄ (1, 1) 1
−2σ3−σ5
−3 6 e+ −12 3
ψ2̄3̄4̄ (1, 3) + (1, 1) 4
−σ2−σ3−σ4
3 0 ν̄ 0 ν̄ 0
ψ2̄3̄5̄ (1, 3) + (1, 1) 4
−σ2−σ3−σ5
−3 6 e+ −6 e− 3
ψ2̄4̄5̄ (1, 2) 2
−σ2−σ4−σ5
−3 l 9 −3 l 3
ψ3̄4̄5̄ (1, 2) 2
−σ3−σ4−σ5
−3 l 9 −9 3
Table 10: The states in the 84 for the SU (3)× (SU (2))2 × (U (1))4 case.
one another, I calculated several numerical properties of each solution, to see if they
distinguished between the solutions. Specifically, I made an arbitrary, but fixed, choice
of one of each charge conjugate pair of (1, 2) states, to include in the tests, and an
arbitrary, but fixed, choice of one of each charge conjugate pair of (1, 1) states, not in
the SU (3)× (SU (2))2× (U (1))4 subgroup, to include in the tests, and then calculated
N , the number of distinct possible choices of a set of q, u, d, l, and e states with the
correct relative Y values, and NB, the number of distinct possible choices of a set of
q, u, d, l, and e states with the correct relative Y values, that admit the existence
of a U (1)B coupling to their baryon number. And, defining integer-valued charges,
for these integer-valued (σ1, σ2, σ3, σ4, σ5), by the numerators in the fourth columns
of Tables 9 and 10, I calculated nq, the number of (3, 2) states with Y = 1; xq, the
number of (3, 2) states with Y = −1; nu, the number of (3, 1) states with Y = 4; xu,
the number of (3, 1) states with Y = −4; nd, the number of (3, 1) states with Y = −2;
xd, the number of (3, 1) states with Y = 2; nl, the number of (1, 2) states tested with
Y = ±3; ne, the number of (1, 1) states tested with Y = ±6; nν , the number of (1, 1)
states tested with Y = 0; and ndiag, the number of (σ2 − σ3), (−σ2 − σ3 − σ4), and
(−σ2 − σ3 − σ5) that are zero.
The result was that the thirteen solutions fell into three groups, with all these
numerical quantities, and also θ2, having the same values, for all the members of
each group. Thus it seems possible that there might be just three physically distinct
solutions, one from each group. I have tabulated the Y values for one representative
solution from each group, in Tables 9 and 10.
The solutions in the first group are (σ1, σ2, σ3, σ4, σ5) = (0,−1,−1,−1, 5),
(1, 0,−3, 0, 3), (−2, 0, 0, 3, 3), (−1, 1,−2, 1, 4), and (2,−2,−2, 1, 1). They all have N =
48, NB = 20, θ
2 = 60, nq = 4, xq = 0, nu = 1, xu = 0, nd = 3, xd = 0, nl = 4, ne = 1,
nν = 2, and ndiag = 1. The Y values for (0,−1,−1,−1, 5) are tabulated in Tables 9
and 10 as Y1.
The solutions in the second group are (σ1, σ2, σ3, σ4, σ5) = (2, 1, 1,−2,−8),
(−2, 3, 3, 0,−6), and (0,−1, 5,−4,−4). They all have N = 400, NB = 208, θ2 = 168,
nq = 4, xq = 0, nu = 5, xu = 0, nd = 1, xd = 0, nl = 4, ne = 5, nν = 4, and ndiag = 2.
The Y values for (2, 1, 1,−2,−8) are tabulated in Tables 9 and 10 as Y2.
The solutions in the third group are (σ1, σ2, σ3, σ4, σ5) = (−2,−3, 3, 0, 6),
(0,−1,−1,−4, 8), (−4, 1, 1, 4, 4), (2, 1,−5,−2, 4), and (4,−3,−3, 0, 0). They all have
N = 1296, NB = 592, θ
2 = 168, nq = 4, xq = 0, nu = 3, xu = 0, nd = 3, xd = 0,
nl = 6, ne = 6, nν = 2, and ndiag = 1. The Y values for (−2,−3, 3, 0, 6) are tabulated
in Tables 9 and 10 as Y3. For this example, we can choose 3B = (1, 0, 0, 0,−3), which
gives the correct baryon number, except for the first four states of the 84, and states
involving σ5, other than the fifth state of the 84. The B values for this choice of B are
tabulated in Tables 9 and 10 as B3.
To determine sin2 θW at unification, for the three groups of models, we recall that
the Standard Model Higgs field is a (1, 2) state of SU (3) × SU (2)diag, with Y = 1.
In the examples in Tables 9 and 10, this could, for example, be an extra-dimensional
component AA245 of AU245, for the Y1 case, AA24̄ of AU24̄, for the Y2 case, and AA34̄
of AU34̄, for the Y3 case. Denoting this field by φi, we find, by the methods of the
preceding subsection, that for the Y2 and Y3 cases, its covariant derivative, times i,
becomes, at low energies:
i∂µφi + g
(σa)ik φk + g
Aµ4φi (558)
where Aµ4 is the gauge field of U (1)Y , in a notation similar to the previous subsection,
and Bµa =
(Aµ2a + Aµ3a) is the gauge field of SU (2)diag. While from equation (117)
of Rosner’s review of the Standard Model [44], the standard covariant derivative, times
i, on the Standard Model Higgs field is:
i∂µφi + gRosnerBµa
(σa)ik φk + g
Rosner
Aµ4φi (559)
Now, as noted above, the Standard Model Higgs field has Y = 1. Thus we see that,
for the Y2 and Y3 cases:
gRosner = g
, g′Rosner = g
(560)
And by definition, tan θW =
Rosner
gRosner
. Hence we find that, for the Y2 and Y3 cases,
sin2 θW =
= 0.300 at unification, which is roughly halfway between the value ≃ 0.23
observed at mZ , and the value
= 0.375 found in conventional grand unification [404],
and reasonably consistent with the unification of the SU (3) and SU (2)diag coupling
constants at around 145 TeV, in the absence of accelerated unification. On the other
hand, for the Y1 case, the
168, in (558) and (560), gets replaced by
60, which gives
sin2 θW =
≃ 0.545 at unification, so the Y1 case does not seem very likely.
Let us now suppose that we have found a smooth compact quotient of CH3 or
H6, and a set of Hodge - de Rham harmonic two-forms embedded in the E8 Cartan
subalgebra as above, such that the net number of chiral zero modes of each of the
left-handed states of one generation of the Standard Model, as in Table 6, is three, and
the net number of chiral zero modes of each fermion state not in the Standard Model,
is zero, and that SU (2)
can be broken to SU (2)diag, in a topologically stabilized
manner, by a “monopole”, embedded in E8 in one or more of whichever of the states
ψ23̄, ψ2̄3̄4̄, and ψ2̄3̄5̄ have vanishing U (1)Y charge, in the example under consideration,
without spoiling this. Then it seems reasonable to expect that the Hodge - de Rham
harmonic two-forms will lead to masses ∼ a TeV for all chiral zero modes that can
be matched in left-handed and right-handed pairs, so that the only light fermions will
be the three generations of Standard Model fermions, except possibly for one or more
light singlet neutrino states, which could obtain very small masses by the generalized
seesaw mechanism to be discussed in subsection 5.7.
Let us now consider an arbitrary proton decay process, proceeding via a dimension 6
term in the Standard Model effective action, such as qqql
cdcucec
cucqq
, or d
cucql
[406],
with the SU (3) and SU (2)diag indices contracted in an appropriate manner, where Λ
is an effective cutoff, that determines the size of the term. Then for any four specific
states from Tables 9 and 10, that have nonvanishing amplitudes in those four types
of Standard Model state, the condition for the existence of a U (1)B, that couples as
a nonzero multiple of baryon number, just on those four states, is three homogeneous
linear equations on the four linearly independent σi, so is always satisfied. Thus those
parts of the arguments of Aranda and Carone [52], that depend only on the existence
of such a U (1)B, would seem to suggest that the contribution of those four states,
to the corresponding term in the Standard Model effective action, will be suppressed.
And since this argument applies to all sets of states from Tables 9 and 10, that have
nonvanishing amplitudes in the Standard Model fermion fields in the effective action
term concerned, we expect the same suppression to apply to the overall coefficient of
that term in the effective action, leading to a large value of the effective cutoff Λ, even
though the relevant U (1)B may be different, for different relevant sets of states from
Tables 9 and 10.
Of course, it was not necessary to require that two of the SU (3)× SU (2) singlets,
with independent U (1) charges, have Y = 0, since the unwanted U (1)’s will become
massive by Witten’s Higgs mechanism, provided that none of them is orthogonal to all
the Hodge - de Rham monopoles in the E8 Cartan subalgebra. So additional solutions
might exist, such that the largest number of SU (3) × SU (2) singlets, with linearly
independent U (1) charges, that have Y = 0, is less than two.
5.6 Models where the Abelian Hodge - de Rham monopoles
break E8 to SU (3)× SU (2)× (U (1))5
I shall now consider some models where the Hodge - de Rham harmonic two-forms
break E8 to SU (3) × SU (2) × (U (1))5. An element of the E8 Cartan subalgebra,
and hence of the SU (9) Cartan subalgebra, that can have a vacuum expectation value
without breaking this subgroup of E8, is, in the SU (9) fundamental, a diagonal matrix,
with diagonal matrix elements
(σ1, σ1, σ1, σ2, σ2, σ3, σ4, σ5, σ6) , (561)
such that:
3σ1 + 2σ2 + σ3 + σ4 + σ5 + σ6 = 0 (562)
The states in the 80, omitting the states in the unbroken SU (3)×SU (2)×(U (1))5,
whose U (1) charges vanish identically, are shown in Table 11, and the states in the 84
are shown in Table 12. There are now fifteen SU (3)×SU (2) singlets, plus their complex
conjugates, whose U (1) charges do not vanish identically, and the U (1) charges of five
of these fifteen SU (3) × SU (2) singlets are linearly independent, so there is now a
possibility of raising the masses of all four unwanted U (1)’s as much as required,
without breaking SU (3) × SU (2) × U (1)Y , and without relying on Witten’s Higgs
mechanism, by choosing the σi such that four SU (3)× SU (2) singlets outside the E8
Cartan subalgebra, with linearly independent U (1) charges, all have vanishing U (1)Y
charge, and could thus have vacuum expectation values without breaking SU (3) ×
SU (2)× U (1)Y .
I did a computer search through all 15!
4!11!
= 1365 choices of which four of the fifteen
SU (3) × SU (2) singlets should be set to have Y = 0, to determine which choices led
to the existence of at least one set of q, u, d, l, and e states with the correct Y values,
such that there exists a U (1)B, defined by a different set of σi, whose couplings to at
least one set of these states are proportional to their baryon number, so that there is a
chance of stabilizing the proton by a version of the Aranda-Carone mechanism. There
were only six distinct solutions, four of which are related by permutations of σ3, σ4,
σ5, and σ6. Taking only one of these four, the three solutions are:
3Y1 = (−2, 3, 0, 0, 0, 0) (563)
Block Multiplet Coupling 3Y1 3B1 3Y2 3Y3
ψ12̄ (3,2) 6
σ1−σ2
−5 1 1 q 1 q
ψ13̄ (3,1) 3
σ1−σ3
−2 d 1 4 u 4 u
ψ14̄ (3,1) 3
σ1−σ4
−2 d 1 4 u −2 d
ψ15̄ (3,1) 3
σ1−σ5
−2 d 1 4 u −2 d
ψ16̄ (3,1) 3
σ1−σ6
−2 d 4 4 u −2 d
ψ21̄ (3̄, 2) 6
−σ1+σ2
5 −1 −1 q̄ −1 q̄
ψ31̄ (3̄, 1) 3
−σ1+σ3
2 d̄ −1 −4 ū −4 ū
ψ41̄ (3̄, 1) 3
−σ1+σ4
2 d̄ −1 −4 ū 2 d̄
ψ51̄ (3̄, 1) 3
−σ1+σ5
2 d̄ −1 −4 ū 2 d̄
ψ61̄ (3̄, 1) 3
−σ1+σ6
2 d̄ −4 −4 ū 2 d̄
ψ23̄ (1, 2) 2
σ2−σ3
3 l̄ 0 3 l̄ 3 l̄
ψ24̄ (1, 2) 2
σ2−σ4
3 l̄ 0 3 l̄ −3 l
ψ25̄ (1, 2) 2
σ2−σ5
3 l̄ 0 3 l̄ −3 l
ψ26̄ (1, 2) 2
σ2−σ6
3 l̄ 3 3 l̄ −3 l
ψ34̄ (1, 1) 1
σ3−σ4
0 0 0 −6 e−
ψ35̄ (1, 1) 1
σ3−σ5
0 0 0 −6 e−
ψ36̄ (1, 1) 1
σ3−σ6
0 3 0 −6 e−
ψ45̄ (1, 1) 1
σ4−σ5
0 0 0 0
ψ46̄ (1, 1) 1
σ4−σ6
0 3 0 0
ψ56̄ (1, 1) 1
σ5−σ6
0 3 0 0
ψ32̄ (1, 2) 2
−σ2+σ3
−3 l 0 −3 l −3 l
ψ42̄ (1, 2) 2
−σ2+σ4
−3 l 0 −3 l 3 l̄
ψ52̄ (1, 2) 2
−σ2+σ5
−3 l 0 −3 l 3 l̄
ψ62̄ (1, 2) 2
−σ2+σ6
−3 l −3 −3 l 3 l̄
ψ43̄ (1, 1) 1
−σ3+σ4
0 0 0 6 e+
ψ53̄ (1, 1) 1
−σ3+σ5
0 0 0 6 e+
ψ63̄ (1, 1) 1
−σ3+σ6
0 −3 0 6 e+
ψ54̄ (1, 1) 1
−σ4+σ5
0 0 0 0
ψ64̄ (1, 1) 1
−σ4+σ6
0 −3 0 0
ψ65̄ (1, 1) 1
−σ5+σ6
0 −3 0 0
Table 11: The states in the 80 for the SU (3) × SU (2) × (U (1))5 case, omitting the
states in the SU (3)× SU (2)× (U (1))5 subgroup.
Block Multiplet Coupling 3Y1 3B1 3Y2 3Y3
ψ1̄1̄1̄ (1, 1) 1
6 e+ −3 −6 e− 0
ψ1̄1̄2̄ (3, 2) 6
−2σ1−σ2
1 q −2 −5 1 q
ψ1̄1̄3̄ (3, 1) 3
−2σ1−σ3
4 u −2 −2 d 4 u
ψ1̄1̄4̄ (3, 1) 3
−2σ1−σ4
4 u −2 −2 d −2 d
ψ1̄1̄5̄ (3, 1) 3
−2σ1−σ5
4 u −2 −2 d −2 d
ψ1̄1̄6̄ (3, 1) 3
−2σ1−σ6
4 u 1 −2 d −2 d
ψ1̄2̄2̄ (3̄, 1) 3
−σ1−2σ2
−4 ū −1 −4 ū 2 d̄
ψ1̄3̄4̄ (3̄, 1) 3
−σ1−σ3−σ4
2 d̄ −1 2 d̄ 2 d̄
ψ1̄3̄5̄ (3̄, 1) 3
−σ1−σ3−σ5
2 d̄ −1 2 d̄ 2 d̄
ψ1̄3̄6̄ (3̄, 1) 3
−σ1−σ3−σ6
2 d̄ 2 2 d̄ 2 d̄
ψ1̄4̄5̄ (3̄, 1) 3
−σ1−σ4−σ5
2 d̄ −1 2 d̄ −4 ū
ψ1̄4̄6̄ (3̄, 1) 3
−σ1−σ4−σ6
2 d̄ 2 2 d̄ −4 ū
ψ1̄5̄6̄ (3̄, 1) 3
−σ1−σ5−σ6
2 d̄ 2 2 d̄ −4 ū
ψ1̄2̄3̄ (3̄, 2) 6
−σ1−σ2−σ3
−1 q̄ −1 −1 q̄ 5
ψ1̄2̄4̄ (3̄, 2) 6
−σ1−σ2−σ4
−1 q̄ −1 −1 q̄ −1 q̄
ψ1̄2̄5̄ (3̄, 2) 6
−σ1−σ2−σ5
−1 q̄ −1 −1 q̄ −1 q̄
ψ1̄2̄6̄ (3̄, 2) 6
−σ1−σ2−σ6
−1 q̄ 2 −1 q̄ −1 q̄
ψ2̄2̄3̄ (1, 1) 1
−2σ2−σ3
−6 e− 0 0 6 e+
ψ2̄2̄4̄ (1, 1) 1
−2σ2−σ4
−6 e− 0 0 0
ψ2̄2̄5̄ (1, 1) 1
−2σ2−σ5
−6 e− 0 0 0
ψ2̄2̄6̄ (1, 1) 1
−2σ2−σ6
−6 e− 3 0 0
ψ2̄3̄4̄ (1, 2) 2
−σ2−σ3−σ4
−3 l 0 3 l̄ 3 l̄
ψ2̄3̄5̄ (1, 2) 2
−σ2−σ3−σ5
−3 l 0 3 l̄ 3 l̄
ψ2̄3̄6̄ (1, 2) 2
−σ2−σ3−σ6
−3 l 3 3 l̄ 3 l̄
ψ2̄4̄5̄ (1, 2) 2
−σ2−σ4−σ5
−3 l 0 3 l̄ −3 l
ψ2̄4̄6̄ (1, 2) 2
−σ2−σ4−σ6
−3 l 3 3 l̄ −3 l
ψ2̄5̄6̄ (1, 2) 2
−σ2−σ5−σ6
−3 l 3 3 l̄ −3 l
ψ3̄4̄5̄ (1, 1) 1
−σ3−σ4−σ5
0 0 6 e+ 0
ψ3̄4̄6̄ (1, 1) 1
−σ3−σ4−σ6
0 3 6 e+ 0
ψ3̄5̄6̄ (1, 1) 1
−σ3−σ5−σ6
0 3 6 e+ 0
ψ4̄5̄6̄ (1, 1) 1
−σ4−σ5−σ6
0 3 6 e+ −6 e−
Table 12: The states in the 84 for the SU (3)× SU (2)× (U (1))5 case.
q q̄ u ū d d̄ l l̄ e+ e− ν q5 q̄5 8 3 Y total
Y1 in 80 0 0 0 0 4 4 4 4 0 0 12 1 1 1 1 5 80
Y1 in 84 1 4 4 1 0 6 6 0 1 4 4 0 0 0 0 0 84
Y1 in 84 4 1 1 4 6 0 0 6 4 1 4 0 0 0 0 0 84
Y1 total 5 5 5 5 10 10 10 10 5 5 20 1 1 1 1 5 248
Y2 in 80 1 1 4 4 0 0 4 4 0 0 12 0 0 1 1 5 80
Y2 in 84 0 4 0 1 4 6 0 6 4 1 4 0 1 0 0 0 84
Y2 in 84 4 0 1 0 6 4 6 0 1 4 4 1 0 0 0 0 84
Y2 total 5 5 5 5 10 10 10 10 5 5 20 1 1 1 1 5 248
Y3 in 80 1 1 1 1 3 3 4 4 3 3 6 0 0 1 1 5 80
Y3 in 84 1 3 1 3 3 4 3 3 1 1 7 0 1 0 0 0 84
Y3 in 84 3 1 3 1 4 3 3 3 1 1 7 1 0 0 0 0 84
Y3 total 5 5 5 5 10 10 10 10 5 5 20 1 1 1 1 5 248
Table 13: The numbers of each type of state, for the three choices of Y in the SU (3)×
SU (2)× (U (1))5 case.
3Y2 = (2, 1,−2,−2,−2,−2) (564)
3Y3 = (0,−1,−4, 2, 2, 2) (565)
All three of these have θ2 = 60, so by the same method as in the previous two subsec-
tions, we find sin2 θW =
at unification, as for SU(5) grand unification, so unification
depends entirely on the accelerated unification mechanism [129, 129, 373]. For Y1, we
could choose 3B = (1, 0, 0, 0, 0,−3), and the resulting values of B are tabulated as B1
in Tables 11 and 12.
The number of states of each type, for each of the three choices of Y , are given in
Table 13. The total number of states of each type is the same for all three choices, so
it seems possible that the three different choices of Y might be physically equivalent.
We note that all the fermion states in the E8 fundamental, that are not in the
SU (3) × SU (2) × (U (1))5 subgroup, and can thus be given a nonzero net number
of chiral zero modes by the Hodge - de Rham harmonic two-forms in the E8 Cartan
subalgebra, are now either Standard Model fermions, as in Table 6, or singlet neutrinos,
apart from the single q5 state with Y = −5
, and the single q̄5 state with Y = 5
is well known that the possible sets of left-handed chiral fermions, in four dimensions,
are very strongly constrained by the requirement of the absence of anomalies [407,
408, 409, 410, 411, 412, 413], and we will now find that an arbitrary set of Hodge -
de Rham harmonic two-forms, of a smooth compact quotient of CH3 or H6 that is a
spin manifold, embedded in the E8 Cartan subalgebra as above, such that Witten’s
topological constraint is satisfied, will result in a set of chiral zero modes that is simply
a number of Standard Model generations.
If the net numbers of left-handed chiral zero modes are nq q’s, nu u’s, nd d’s, nl l’s,
ne e’s, and n5 q
5’s, then the conditions for the absence of gauge anomalies [407, 408],
and mixed gauge-gravitational anomalies [414, 415, 416], in four dimensions, are as
follows.
From a triangle diagram with three external SU (3) gauge bosons:
2nq + nu + nd + 2n5 = 0 (566)
From a triangle diagram with two external SU (3) gauge bosons, and one external
U (1)Y gauge boson:
nq + 2nu − nd − 5n5 = 0 (567)
From a triangle diagram with two external SU (2) gauge bosons, and one external
U (1)Y gauge boson:
nq − nl − 5n5 = 0 (568)
From a triangle diagram with three external U (1)Y gauge bosons:
nq + 32nu − 4nd − 9nl + 36ne − 125n5 = 0 (569)
And from a triangle diagram with two external gravitons, and one external U (1)Y
gauge boson [414, 415, 416]:
nq + 2nu − nd − nl + ne − 5n5 = 0 (570)
The five equations (566), (567), (568), (569), and (570), are linearly independent,
and the general solution, with integer values for the ni, is an integer multiple of one
Standard Model generation, which has (nq, nu, nd, nl, ne, n5) = (1,−1,−1, 1, 1, 0). Thus
for an arbitrary set of Hodge - de Rham harmonic two-forms, in the Cartan subalgebra
of E8, that break E8 to SU (3)× SU (2)×U (1)Y as considered in this subsection, and
satisfy Witten’s topological constraint, the chiral fermions will consist of an integer
number of Standard Model generations.
Let us now consider the case where U (1)Y is (−2,−2,−2, 3, 3, 0, 0, 0, 0). To ensure
that U (1)Y does not get a mass by Witten’s Higgs mechanism [45], the Abelian vacuum
gauge fields (σ1, σ1, σ1, σ2, σ2, σ3, σ4, σ5, σ6) with non-vanishing field strength must be
perpendicular to U (1)Y . But this implies that σ1 = σ2, so that the Abelian vacuum
gauge fields with non-vanishing field strength must actually leave SU (5) unbroken.
Nevertheless, we would still be able to break E8 to the Standard Model by topologically
stabilized vacuum gauge fields in the E8 Cartan subalgebra, if we could topologically
stabilize a Hosotani vacuum gauge field with vanishing field strength [49, 50, 51] that
is in the Cartan subalgebra but not perpendicular to U (1)Y . This could be achieved if
the fundamental group of the compact six-manifold M6 included a non-trivial element
a such that an = 1 for some finite integer n, because an Abelian Wilson line looping
once round the closed path corresponding to a will then be a phase factor f satisfying
fn = 1. But as noted in section 3, on page 160, the smooth compact quotients M6
considered in this paper have no such non-trivial elements a, called torsion elements.
Nevertheless, examples in three dimensions show that it is possible for the first
homology groupH1 (M,Z) of a hyperbolic manifoldM to have torsion even though the
fundamental group of M has no torsion. For example, using Weeks’s program SnapPea
[312], the Weeks manifold, which is the compact hyperbolic three-manifold of smallest
known volume, and designated m003(-3,1) by SnapPea, is found to have first homology
group Z/5+Z/5. This can be checked using the presentation of the fundamental group
given by SnapPea, which has generators a, b, and relations a2b2a2b−1ab−1 = 1 and
a2b2a−1ba−1b2 = 1. We obtain the first homology group from the fundamental group
by treating the generators as commuting in the relations, which then collapse to a5 = 1
and b5 = 1. SnapPea also confirms that the Weeks manifold is oriented.
Thus it seems reasonable to expect that there may exist smooth compact quotients
M6 of CH3 or H6 such that H1 (M6,Z) has torsion. This would be sufficient to
obtain a topologically stabilized Hosotani Abelian vacuum gauge field with vanishing
field strength, even though the fundamental group of M6 has no torsion. For suppose
there exists a one-cycle l that is not a boundary, such that nl, for some finite integer
n, is a boundary. We consider an E8 Wilson line w that loops once around l. Suppose
there is a Hosotani U (1) vacuum field that is locally pure gauge, but for which w is a
non-trivial phase factor. Then wn is a phase factor along a one-cycle that is a boundary.
Thus since the Hosotani field is locally pure gauge, we find wn = 1 by Stokes’s theorem.
As in the preceding section, it seems reasonable to expect that the Hodge - de Rham
harmonic two-forms will lead to masses ∼ a TeV for all chiral zero modes that can be
matched in left-handed and right-handed pairs, so that the only light fermions will
be the three generations of Standard Model fermions, except possibly for one or more
light singlet neutrino states, which could obtain very small masses by the generalized
seesaw mechanism to be discussed in the following subsection.
And as in the preceding subsection, let us now consider an arbitrary proton decay
process, proceeding via a dimension 6 term in the Standard Model effective action, such
as qqql
cdcucec
cucqq
, or d
cucql
[406], with the SU (3) and SU (2)diag indices contracted
in an appropriate manner, where Λ is an effective cutoff, that determines the size of the
term. Then for any four specific states from Tables 11 and 12, that have nonvanishing
amplitudes in those four types of Standard Model state, the condition for the existence
of a U (1)B, that couples as a nonzero multiple of baryon number, just on those four
states, is three homogeneous linear equations on the five linearly independent σi, so
is always satisfied. Thus those parts of the arguments of Aranda and Carone [52],
that depend only on the existence of such a U (1)B, would seem to suggest that the
contribution of those four states, to the corresponding term in the Standard Model
effective action, will be suppressed. And since this argument applies to all sets of
states from Tables 11 and 12, that have nonvanishing amplitudes in the Standard Model
fermion fields in the effective action term concerned, we expect the same suppression
to apply to the overall coefficient of that term in the effective action, leading to a large
value of the effective cutoff Λ, even though the relevant U (1)B may be different, for
different relevant sets of states from Tables 11 and 12.
To find out whether the mass hierarchy of the observed quarks and charged leptons
could occur by a version of the Arkani-Hamed - Schmaltz mechanism [390], in the type
of model considered here, it would be necessary to find the explicit form of the Hodge
- de Rham harmonic two-forms, for examples of smooth compact quotients of CH3
or H6 that are spin manifolds, and the corresponding chiral fermion zero modes, to
find out how spread out or localized they are. However, we note that in the examples
considered by Arkani-Hamed and Schmaltz [390], and by Acharya and Witten [417],
the chiral fermion modes have a Gaussian shape, even though the fermion “mass terms”
only depend linearly on position. The explicit forms of the chiral fermion zero modes in
monopole backgrounds on the two-sphere have been given by Deguchi and Kitsukawa
[418].
Of course, it was not necessary to require that four of the SU (3)× SU (2) singlets,
with linearly independent U (1) charges, have Y = 0, since the unwanted U (1)’s will
become massive by Witten’s Higgs mechanism, provided that none of them is orthogo-
nal to all the Hodge - de Rham monopoles in the E8 Cartan subalgebra. So additional
solutions might exist, such that the largest number of SU (3) × SU (2) singlets, with
linearly independent U (1) charges, that have Y = 0, is three or less.
5.7 Generalized seesaw mechanism
With regard to how small neutrino masses, ∼ 1 eV or smaller, might arise in models
of the type considered in this paper, it seems to be possible to produce a very small,
but nonzero, eigenvalue, from a matrix whose matrix elements are integers in the range
0 to 10, if all the matrix elements in the lower right triangle, below the lower left to
upper right diagonal, are zero, the matrix elements on the lower left to upper right
diagonal are 1, and the matrix elements in the upper left triangle, above the lower left
to upper right diagonal, are ∼ 10. For example the matrix:
10 10 10 1
10 10 1 0
10 1 0 0
1 0 0 0
(571)
has eigenvalues 22.891, −7.6024, 4.7127, and −0.001219. If this effect occurs because
all but one of the eigenvalues tend to be comparable to the large matrix elements in
the upper left triangle, but the determinant, and hence the product of the eigenvalues,
is equal to 1, it would presumably be possible to obtain an eigenvalue as small as
required, by considering larger matrices with this structure. We note that in the
models considered in the preceding subsection, it might be natural to find a number
∼ 10 or more of singlet neutrinos, which could perhaps sometimes have a mass matrix
of this type. To obtain the required small eigenvalue, the matrix elements in the lower
right triangle would presumably have to be exactly zero. This would presumably be
possible, if the matrix elements were integer multiples of an overall factor, but I do not
know of a reason why this should be so.
Acknowledgements
I would like to thank Savas Dimopoulos, David E. Kaplan, and Karin Slinger for or-
ganizing a very enjoyable and helpful visit to Stanford University ITP, where part of
the work that led to this paper was carried out, Nima Arkani-Hamed, Savas Dimopou-
los, Michal Fabinger, Simeon Hellerman, Veronika Hubeny, Shamit Kachru, Nemanja
Kaloper, Renata Kallosh, David E. Kaplan, Matt Kleban, Albion Lawrence, Andrei
Linde, John McGreevy, Michael Peskin, Steve Shenker, Eva Silverstein, Matt Strassler,
and Lenny Susskind for helpful discussions or comments, and Fyodor Tkachov and
Kasper Peeters for helpful emails.
At an early stage of this work, some of the calculations were carried out using
TeXaide [419] and TeXnic Center [420], rather than by using pen and paper. Subse-
quently, after migrating to Debian GNU/Linux [421], the work was done almost entirely
by means of GNU TeXmacs [422], without using pen and paper at all. The paper was
largely written using GNU TeXmacs, and ported to KTeXmaker2, now renamed to Kile
[423], for completion. Some of the calculations were done with PARI/GP [274] and
Maxima [291], initially freestanding, and subsequently run from within GNU TeXmacs.
The LATEX pictures were prepared with TexPict [424]. The online Wolfram Integrator
[425] was used to perform some integrals. The bibliography of version 2 was sequenced
with help from Ordercite [426].
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	1 Introduction
	2 Thick pipe geometries
	2.1 Horava-Witten theory
	2.2 The complex hyperbolic space  CH^3 
	2.3 The field equations and boundary conditions
	2.3.1 The Christoffel symbols, Riemann tensor, and Ricci tensor
	2.3.2 The Yang-Mills coupling constants in four dimensions
	2.3.3 The problem of the higher order corrections to Horava-Witten theory
	2.3.4 The Casimir energy density corrections to the energy-momentum tensor
	2.3.5 The orders of perturbation theory that the terms in the Casimir energy densities occur at
	2.3.6 The expansion parameter
	2.3.7 Witten's topological constraint
	2.3.8 The field equations and boundary conditions for the three-form  gauge field
	2.3.9 The field equations and boundary conditions for the metric
	2.4 Analysis of the Einstein equations and the boundary conditions for the metric
	2.4.1 Beyond the proximity force approximation
	2.4.2 The region near the inner surface of the thick pipe
	2.4.3 The boundary conditions at the inner surface of the thick pipe
	2.4.4 The classical solutions in the bulk
	2.5 Solutions with both a and b large compared to ^2/9, at the outer surface of the thick pipe
	2.5.1 Newton's constant and the cosmological constant for solutions with the outer surface in the classical region
	2.6 Solutions with a as small as ^2/9, at the outer surface of the thick pipe
	2.6.1 Newton's constant and the cosmological constant
	2.6.2 Comparison with sub-millimetre tests of Newton's law
	2.6.3 Comparison with precision solar system tests of General Relativity
	2.6.4 Further consequences of the warp factor decreasing to a small value, at the outer surface of the thick pipe
	2.7 Stiffening by fluxes wrapping three-cycles of the compact six-manifold times the radial dimension
	2.7.1 The region near the outer surface
	2.7.2 Newton's constant and the cosmological constant in the presence of the extra fluxes
	3 Smooth compact quotients of  CH^3 ,  H^6 ,  H^3  and  S^3 
	3.1 Smooth compact arithmetic quotients of C H^n and H^n
	3.1.1 Compactness of G /  for the examples of arithmetic lattices
	3.1.2 Obtaining finite index torsion-free subgroups of  by Selberg's lemma
	3.2 Smooth compact quotients of S^3
	4 The Casimir energy densities
	4.1 The Salam-Strathee harmonic expansion method
	4.2 AdS_5 C P^3 compactification of d = 11 supergravity
	5  E8  vacuum gauge fields and the Standard Model
	5.1 The lightest massive modes of the supergravity multiplet
	5.2 An SU ( 9 ) basis for E_8
	5.3 Dirac quantization condition for  E8  vacuum gauge fields
	5.4 Nonexistence of models where the Abelian Hodge - de Rham monopoles break E_8 to SU ( 3 ) ( SU ( 2 ) )^3 ( U ( 1 ) )^3
	5.5 Models where the Abelian Hodge - de Rham monopoles break E_8 to SU ( 3 ) ( SU ( 2 ) )^2 ( U ( 1 ) )^4
	5.6 Models where the Abelian Hodge - de Rham monopoles break E_8 to SU ( 3 ) SU ( 2 ) ( U ( 1 ) )^5
	5.7 Generalized seesaw mechanism
