Exact Solutions of Einstein-Yang-Mills Theory with Higher-Derivative
  Coupling
OCU-PHYS 263
Exact Solutions of Einstein-Yang-Mills Theory with
Higher-Derivative Coupling
Hironobu Kihara∗
Osaka City University, Advanced Mathematical Institute (OCAMI),
3-3-138 Sugimoto, Sumiyoshi, Osaka 558-8585, Japan
Muneto Nitta†
Department of Physics, Keio University, Hiyoshi,
Yokohama, Kanagawa 223-8521, Japan
Abstract
We construct a classical solution of an Einstein-Yang-Mills system with a fourth order term
with respect to the field strength of the Yang-Mills field. The solution provides a compactification
proposed by Cremmer and Scherk; ten-dimensional space-time with a cosmological constant is
compactified to the four-dimensional Minkowski space with a six-dimensional sphere S6 on which
an instanton solution exists. The radius of the sphere is not a modulus but is determined by the
gauge coupling and the four-derivative coupling constants and the Newton’s constant. We also
construct a solution of ten-dimensional theory without a cosmological constant compactified to
AdS4 × S6.
∗Electronic address: kihara(at)sci.osaka-cu.ac.jp
†Electronic address: nitta(at)phys-h.keio.ac.jp
http://arxiv.org/abs/0704.0505v3
mailto:kihara(at)sci.osaka-cu.ac.jp
mailto:nitta(at)phys-h.keio.ac.jp
Unification of fundamental forces with space-time and matter often requires higher-
dimensional space-time rather than our four-dimensional Universe. The early Kaluza-Klein
theory unifies gravity and the electro-magnetic interaction by considering five-dimensional
space-time with one direction compactified into a circle S1 [1]. This old idea has been revis-
ited several times. After supergravity was discovered many people tried to unify all forces
and matter in higher-dimensional space-time compactified on various internal manifolds [2].
String theory was proposed as the most attractive candidate of unification, but it is defined
only in ten-dimensional space-time. In order to realize four-dimensional Universe one has
to find a suitable six-dimensional internal space. So many candidates of such spaces were
proposed; Calabi-Yau manifolds and orbifold models. Internal manifolds can be deformed
with satisfying the Einstein equation, and these degrees of freedom are called the moduli.
The moduli introduce unwanted massless particles in four-dimensional world. Recently a
new mechanism has been suggested to fix these moduli by turning on the Ramond-Ramond
flux on the internal space [3, 4]. This flux compactification has been extensively studied in
these years.
We would like to revise the compactification scenario with fixed moduli proposed by
Cremmer and Scherk long time ago [5] (see also [6, 7]) in a theory with a cosmological
constant. By placing solitons on a compact internal space they showed decompactifying limit
with large radius of the internal space is disfavored and the radius is fixed to a certain value
determined by coupling constants. They considered the ’t Hooft-Polyakov monopole [8] on
S2 and the Yang-Mills instanton [9] on S4, both of which can satisfy, with proper coupling
constants, the first order (self-dual) equations rather than the second order equations of
motion, but their solutions on higher dimensional sphere are not the case. Since string
theory is defined in ten dimensions, it is natural to consider this scenario with stable BPS
solitons on a six-dimensional internal space like S6.
Higher dimensional generalization of self-dual equations was suggested by Tchrakian some
years ago [10]. Eight dimensional case is known as octonionic instantons [11]. Though several
works have been done for generalized self-dual equations [12, 13], a six-dimensional case was
not discussed because of the lack of conformal property. Recently we have found a new
solution to the generalized self-dual equations in an SO(6) pure Yang-Mills theory with a
fourth order term with respect to the field strength of the Yang-Mills field (a four-derivative
term) on a six-dimensional sphere S6 [14].
In this letter we propose to use this solution in the context of a compactification of the
Cremmer-Scherk type. In our model ten-dimensional space-time with (without) a cosmo-
logical constant is compactified to a four-dimensional Minkowski space M4 (anti de Sitter
space AdS4) with a six-dimensional sphere S
6, where dimensionality of the internal space,
six, is required by the four derivative term. Unlike the case of the absence of gravity [14] the
four-derivative coupling constant α can differ from the constant β in the generalized self-dual
equations. When the relation α = β holds the generalized self-dual equations become the
Bogomol’nyi equations and solutions are BPS. We find for both M4 × S6 and AdS4 × S6
that certain relations exist between the radius of S6, the gauge coupling, the four-derivative
coupling α and the gravitational coupling constants. When the four-derivative coupling con-
stant α vanishes in the case of M4 × S6, these relations reduce to those of the original work
by Cremmer and Scherk. The advantage of our model to the Cremmer-Scherk model is that
the Yang-Mills soliton in our model satisfies the self-dual equations (the Bogomol’nyi equa-
tions for α = β) rather than usual equations of motion in the case of the Cremmer-Scherk
model. This ensures the stability of configuration at least for the sector of Yang-Mills fields.
Let us consider that space-time is a ten-dimensional manifold. We consider an Einstein-
Yang-Mills theory. Our action contains as dynamical variables the Yang-Mills (gauge) fields
and a graviton field or the metric ĝµ̂ν̂ . Indices with a hat “̂ ” will refer to a ten-
dimensional space-time (X0, X1, · · · , X9). Latin indices (a, b, · · · ) run from 1 to 6 and refer
to an internal space. The Clifford algebra associated with the orthogonal group SO(6) is
useful and we represent generators of the Lie algebra so(6) as their elements. The Clifford
algebra is defined by gamma matrices {Γa} which satisfy the following anti-commutation
relations, {Γa,Γb} = 2δab. These matrices can be realized as 8 × 8 matrices with complex
coefficients. The generators of so(6) are represented by Γab =
[Γa,Γb]. We often abbreviate
the Yang-Mills fields as Aµ̂ =
Γab and we also use notations with differential forms.
Thus the gauge fields are expressed as A = Aµ̂dX
µ̂. In this notation, the corresponding field
strength F is written as F = dA+eA∧A, where e is a gauge coupling. Covariant derivative
Dµ̂ on an adjoint representation Y =
Y [ab]Γab is defined as Dµ̂Y = ∂µ̂Y + e(Aµ̂Y − Y Aµ̂),
where Y is a scalar multiplet. The action Stotal consists of two parts. One is the Einstein-
Hilbert action SE and the other SYMT is a Yang-Mills action with a term which is the fourth
power of the field strength F . Such a quartic term has been studied by Tchrakian [10] and
so we call it the Tchrakian term. The total action is:
Stotal = SE + SYMT , SE =
dvR ,
SYMT =
−F ∧ ∗F + α2(F ∧ F ) ∧ ∗(F ∧ F )− V0dv
. (1)
Here the 10-form dv is an invariant volume form with respect to the metric ĝ and is written
as dv =
−ĝd10X in a local patch. The scalar curvature is denoted by R. The asterisk “∗”
denotes the Hodge dual operator. This operator defines an inner product over differential
forms, and for a given form ω, ω∧∗ω is proportional to the invariant volume form dv.1 The
parameters of this action are the Newton’s gravitational constant G, the gauge coupling e,
the four-derivative coupling α and the cosmological constant V0.
We show the explicit form of the Yang-Mills part with components of A and F ,
SYMT = −
µ̂ν̂ F
µ̂ν̂,[ab] +
[abcd]
µ̂ν̂ρ̂σ̂ T
µ̂ν̂ρ̂σ̂,[ab][cd] +
3 · 16
Sµ̂ν̂ρ̂σ̂S
µ̂ν̂ρ̂σ̂ +
, (2)
dX µ̂ ∧ dX ν̂Γab , Sµ̂ν̂ρ̂σ̂ = F [ab]µ̂ν̂ F
, (3)
[ab][cd]
µ̂ν̂ρ̂σ̂
[abcd]
µ̂ν̂ρ̂σ̂
[ab][cd]
µ̂ν̂ρ̂σ̂
[ac][db]
µ̂ν̂ρ̂σ̂
[ad][bc]
µ̂ν̂ρ̂σ̂
[cd][ab]
µ̂ν̂ρ̂σ̂
[db][ac]
µ̂ν̂ρ̂σ̂
[bc][ad]
µ̂ν̂ρ̂σ̂
. (4)
The Euler-Lagrange equations from these actions read the usual Einstein equation and the
equations for the Yang-Mills fields:
Rµ̂ν̂ −
ĝµ̂ν̂R = 8πGTµ̂ν̂ , Dµ̂
−gF µ̂ν̂ − 2α2
−gF [µ̂ν̂F ρ̂σ̂]Fρ̂σ̂
= 0 . (5)
Here the energy-momentum tensor Tµ̂ν̂ is obtained by the variation of the Yang-Mills part
with respect to the metric:
Tµ̂ν̂ =
ρ̂,[ab]F
+ α2T̃
[abcd]
µ̂ρ̂σ̂τ̂
ρ̂σ̂τ̂ ,[ab][cd] +
3 · 2
Sµ̂ρ̂σ̂τ̂Sν̂
ρ̂σ̂τ̂ − 1
gµ̂ν̂χ
F µ̂ν̂,[ab] +
[abcd]
µ̂ν̂ρ̂σ̂
T µ̂ν̂ρ̂σ̂,[ab][cd] +
3 · 8
Sµ̂ν̂ρ̂σ̂S
µ̂ν̂ρ̂σ̂ + V0 . (6)
To solve these equations, we make an ansatz which is the same as that of Cremmer-Scherk.
Our ansatz for the metric is the following:
ds2 = ηµνdx
µdxν +
(1 + y2/4R20)
dyIdyJ = ĝµ̂ν̂dX
µ̂dX ν̂ , y2 =
(yI)2 , (7)
1 The Hodge dual operator acting on a differential form on a space with Minkowski signature satisfies the
following relation: (Fµνdx
µν) ∧ ∗(Fρσdxρσ) = −FµνFµνdv.
where the coordinates X are the total space-time coordinates. The metric ηµν = diag(− +
++) is the Lorentz metric on the four-dimensional Minkowski space. Greek indices without
a hat “̂ ”, for instance µ will refer to the first four variables. Capital indices (I, J, · · · ) run
from one to six and refer to the compact space. The six-dimensional space is taken as a
sphere with a radius R0. The Riemann tensor, Ricci tensor and scalar curvature are
RIJKL =
δIKgJL − δILgJK
, RIJ =
gIJ , R =
. (8)
The rest components of the curvature tensor vanish. In this space, the Einstein equations
in (5) reduce to simple equations,
= 8πGTµν , 0 = TµI , −
gIJ = 8πGTIJ . (9)
We now make ansatzes for the gauge fields. We assume that the fields A do not depend
on the four-dimensional directions, ∂µA = 0, and they have no four-dimensional components
Aµ = 0. This implies that the field strengths are two forms on the six-dimensional sphere:
A = AI(y)dy
I , F =
FIJdy
I ∧ dyJ . With these ansatzes, the four-dimensional part of the
energy-momentum tensor becomes −1
ηµνχ, and the equation reduces to 30/R
0 = 8πGχ.
This equation requires that the χ is a constant. Suppose that the field strength fulfils the
generalized self-dual condition
F = iβγ7 ∗6 (F ∧ F ), (10)
where β is a real parameter. Here “∗6” means the Hodge dual on the six-dimensional sphere.
Then the second part of the equations of motion (5) is fulfilled automatically by the relation
DF = 0, where the exterior covariant derivative is defined as DF = dF+e (A ∧ F − F ∧A).
In fact we have an explicit solution to the self-dual equation:
4eR20
yaebΓab , F =
4eR20
ea ∧ ebΓab , β =
. (11)
Here we identify the internal space index and the sphere index. The energy-momentum
tensor of this configuration becomes
, χ = (1 + ζ)
4e2R40
+ V0 , TIJ = −
(1− ζ)
4e2R40
gIJ . (12)
With these ansatzes we obtain algebraic equations from the Einstein equations:
= 8πG
(1 + ζ)
2e2R40
= 8πG
(1− ζ) 5
8e2R40
, (13)
From these we finally obtain
e2R20
(2 + 4ζ) , V0 =
4e2R40
(1 + 3ζ) . (14)
When the four-derivative coupling vanishes, α = 0 and therefore ζ = 0, these relations
reduce to those of the Cremmer and Scherk [5]. 2 When the relation α = β holds (ζ = 1)
our solution saturates the Bogomol’nyi bound and becomes a BPS state. The energy density
is given by an integral over S6 as follows:
YMT =
−F ∧ ∗6F + α2(F ∧ F ) ∧ ∗6(F ∧ F )
Tr (iF ∓ αγ7 ∗6 (F ∧ F )) ∧ ∗6 (iF ∓ αγ7 ∗6 (F ∧ F ))±
Trγ7F ∧ F ∧ F
≥ ± i
Trγ7F ∧ F ∧ F = ∓
ǫabcdefF
[ab] ∧ F [cd] ∧ F [ef ] ≡ ±Q , (15)
where the field strength F has only components along S6. If the coupling α is equal to β,
the solution of eq. (10) satisfies the Bogomol’nyi equation and the energy attains the local
minimum. We can also consider a system coupled with scalar fields. Suppose that scalar
fields Qm transform as a representation of SO(6). The index m labels the representation
space. Let us add an action SQ of the scalar fields Q with a Higgs potential
dvDµ̂Q
mDµ̂Qm + V (Q2)
, Dµ̂Q
m = ∂µ̂Q
R(Γab)mm′Q
m′ (16)
to Stotal. The equations of motion are modified. In general, our solution mentioned above
does not satisfy the modified equations any more. However, for the scalars which fulfil the
covariantly constant condition Dµ̂Q
m = 0 and attain the absolute minimum V (Q) = 0,
the configurations of A and g in equations (7), (11) are still solutions for the modified
equations. Here the constant value of the minimum is shifted to 0. Thus we can argue the
Higgs mechanism around our solutions.
Next we suppose that the four-dimensional part is an anti-de Sitter space AdS4 of radius
RA. Our ansatz for the metric is the following:
ds2 = ηµν(x)dx
µdxν + gIJ(y)dy
IdyJ = ĝµ̂ν̂dX
µ̂dX ν̂ , (17)
2 We need to redefine e the half when we compare to the result of [5].
gIJ(y)dy
IdyJ =
(1 + y2/4R20)
dyIdyJ y2 =
(yI)2 ,
ηµν(x)dx
µdxν =
cos2 θ
−dτ 2 + dθ2 + sin2 θdΩ2
, dΩ2 =
|dz|2
(1 + |z|2/4)2
, (18)
where z parametrizes a whole complex plane. The metric ηµν(x) is a maximally symmetric
metric on the four-dimensional anti-de Sitter space. The Riemann tensor and the Ricci
tensor are
Rµνρσ = −
δµρηνσ − δµσηνρ
, Rµν = −
ηµν ,
RIJKL =
δIKgJL − δILgJK
, RIJ =
gIJ . (19)
The total scalar curvature is obtained by a summation of those of two parts: R = −
In this space, the Einstein equations are
Rµν −
ηµνR = 8πGTµν , RIJ −
gIJR = 8πGTIJ . (20)
The ansatz for the gauge fields is the same as previous one and the energy momentum tensor
does not change. With these ansatzes, we obtain algebraic equations from the Einstein
equations as
= −4πG
(1 + ζ)
4e2R40
= −4πG
(1− ζ) 5
4e2R40
We are interested in a possible relation to string theory and therefore we consider the case
with the vanishing cosmological constant, V0 = 0. In this case, the radii (RA, R0) are
written by the couplings,
R20 = (5 + 7ζ)
, R2A =
5 + 7ζ
5 + 15ζ
R20. (22)
Thus the additional higher derivative coupling term of the Tchrakian type does not affect
critically to the equations of motion. When ζ = 1 our solution becomes a solution of the
Bogomol’nyi equation again.
Our solutions introduced in this letter are new solutions of the system with a Tchrakian
term. The origin of this term has not been clear so far but it seems rather universal in order
to construct solitons with codimensions higher than four: for instance it has played a crucial
role to construct a finite energy monopole (with codimension five) in a six-dimensional space-
time [13]. Though the parameter ζ(= α2/β2) is a free parameter, we expect that the system
goes to ζ = 1 because it becomes BPS. There are several discussions on the (in)stability of
higher-dimensional Yang-Mills theories [15]. To compute the mass spectra of the fluctuations
around our solutions is a future work. When the scalar fields Qm are non-trivially coupled,
the system may allow BPS composite solitons which are made of solitons with different
codimensions, as in the case of usual self-dual Yang-Mills equations coupled to Higgs fields
[16].
Finally our solution of AdS4 ×S6 may have a relation with D2-branes, and we hope that
there exists some impact on AdS/CFT duality [17].
Acknowledgments
We are grateful to D. H. Tchrakian for various comments. We would like to thank
Y. Hosotani, H. Itoyama, Y. Yasui, M. Sakaguchi, T. Oota, T. Kimura, S. Shimasaki and
E. Itou. We also thank M. Sheikh-Jabbari for an advice. This work is supported by the
21 COE program “Constitution of wide-angle mathematical basis focused on knots” from
Japan Ministry of Education.
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	Acknowledgments
	References
