Cosmological Singularities and a Conjectured Gravity/Coset
  Correspondence
arXiv:0704.0732v1  [hep-th]  5 Apr 2007
Cosmological Singularities and a Conjectured
Gravity/Coset Correspondence1
Thibault Damour
Institut des Hautes Etudes Scientifiques, 35 route de Chartres,
F-91440 Bures-sur-Yvette, France
Abstract: We review the recently discovered connection be-
tween the Belinsky-Khalatnikov-Lifshitz-like “chaotic” structure of
generic cosmological singularities in eleven-dimensional supergrav-
ity and the “last” hyperbolic Kac-Moody algebra E10. This intrigu-
ing connection suggests the existence of a hidden “correspondence”
between supergravity (or even M-theory) and null geodesic motion
on the infinite-dimensional coset space E10/K(E10). If true, this
gravity/coset correspondence would offer a new view of the (quan-
tum) fate of space (and matter) at cosmological singularities.
1 Introduction
It is a pleasure to participate in the celebration of the seminal accomplish-
ments of Gabriele Veneziano. I will try to do so by reviewing a line of
research which is intimately connected with several of Gabriele’s important
contributions, being concerned with the cardinal problem of String Cosmol-
ogy: the fate of the Einstein-like space-time description at big crunch/big
bang cosmological singularities. Actually, the work, described below started
as a by-product of the string cosmology program initiated by M. Gasperini
1Contribution to “String Theory and Fundamental Interactions” – in celebration of
Gabriele Veneziano’s 65th birthday – eds. M. Gasperini and J. Maharana, Springer-Verlag,
Heidelberg, 2007.
http://arxiv.org/abs/0704.0732v1
and G. Veneziano [1]. While collaborating with Gabriele on the possible
birth of “pre-big bang bubbles” from the gravitational-collapse instability of
a generic string vacuum made of a stochastic bath of incoming gravitational
and dilatonic waves [2], an issue raised itself : what is the structure of a
generic spacelike (i.e. big crunch or big bang) singularity within the effec-
tive field theory approximation of (super-) string theory (when keeping all
fields, and not only the metric and the dilaton). The answer turned out to be
surprisingly complex, and rich of hidden structures. It was first found [3, 4]
that the general solution, near a spacelike singularity, of the massless bosonic
sector of all superstring models (D = 10, IIA, IIB, I, HE, HO), as well as
that of M theory (D = 11 supergravity), exhibits a never ending oscillatory
behaviour of the Belinsky-Khalatnikov-Lifshitz (BKL) type [5]. However, it
was later realized that behind this seeming entirely chaotic behaviour there
was a hidden symmetry structure [6, 7, 8]. This led to the conjecture of the
existence of a hidden equivalence (i.e. a correspondence) between two seem-
ingly very different dynamical systems: on the one hand, 11-dimensional
supergravity (or even, hopefully, “M-theory”), and, on the other hand, a
one-dimensional E10/K(E10) nonlinear σ model, i.e. the geodesic motion of
a massless particle on the infinite-dimensional coset space2 E10/K(E10) [8].
The intuitive hope behind this conjecture is that the BKL-type near space-
like singularity limit might act as a tool for revealing a hidden structure, in
analogy to the much better established AdS/CFT correspondence [9], where
the consideration of the near horizon limit of certain black D-branes has
revealed a hidden equivalence between 10-dimensional string theory in AdS
spacetime on one side, and a lower-dimensional CFT on the other side. If
the (much less firmly established) “gravity/coset correspondence” were con-
firmed, it might provide both the basis of a new definition of M-theory, and
a description of the “de-emergence” of space near a cosmological singularity
(see [10] and below).
2Here K(E10) denotes the (formal) “maximal compact subgroup” of the hyperbolic
Kac-Moody group E10.
2 Cosmological billiards
Let us start by summarizing the BKL-type analysis of the “near spacelike
singularity limit”, that is, of the asymptotic behaviour of the metric gµν(t,x),
together with the other fields (such as the 3-form Aµνλ(t,x) in supergravity),
near a singular hypersurface. The basic idea is that, near a spacelike singu-
larity, the time derivatives are expected to dominate over spatial derivatives.
More precisely, BKL found that spatial derivatives introduce terms in the
equations of motion for the metric which are similars to the “walls” of a
billiard table [5]. To see this, it is convenient [11] to decompose the D-
dimensional metric gµν into non-dynamical (lapse N , and shift N
i, here set
to zero) and dynamical (e−2β
, θai ) components. They are defined so that the
line element reads
ds2 = −N2dt2 +
θai θ
idxj . (1)
Here d ≡ D − 1 denotes the spatial dimension (d = 10 for SUGRA11, and
d = 9 for string theory), e−2β
represent (in an Iwasawa decomposition)
the “diagonal” components of the spatial metric gij, while the “off diagonal”
components are represented by the θai , defined to be upper triangular matrices
with 1’s on the diagonal (so that, in particular, det θ = 1).
The Hamiltonian constraint, at a given spatial point, reads (with Ñ ≡
det gij denoting the “rescaled lapse”)
H(βa, πa, P, Q)
Gabπaπb +
cA(Q,P, ∂β, ∂
2β, ∂Q) exp
− 2wA(β)
. (2)
Here πa (with a = 1, ..., d) denote the canonical momenta conjugate to the
“logarithmic scale factors” βa, while Q denote the remaining configuration
variables (θai , 3-form components Aijk(t,x) in supergravity), and P their
canonically conjugate momenta (P ia, π
ijk). The symbol ∂ denotes spatial
derivatives. The (inverse) metric Gab in Eq. (2) is the DeWitt “superspace”
metric induced on the β’s by the Einstein-Hilbert action. It endows the
d-dimensional3 β space with a Lorentzian structure Gab β̇
aβ̇b.
One of the crucial features of Eq. (2) is the appearance of Toda-like
exponential potential terms ∝ exp(−2wA(β)), where the wA(β) are linear
forms in the logarithmic scale factors: wA(β) ≡ wAa βa. The range of labels
A and the specific “wall forms” wA(β) that appear depend on the considered
model. For instance, in SUGRA11 there appear: “symmetry wall forms”
wSab(β) ≡ βb − βa (with a < b), “gravitational wall forms” w
abc(β) ≡ 2βa +
e 6=a,b,c
βe (a 6= b, b 6= c, c 6= a), “electric 3-form wall forms”, eabc(β) ≡
βa + βb + βc (a 6= b, b 6= c, c 6= a), and “magnetic 3-form wall forms”,
ma1....a6 ≡ βa1 + βa2 + ...+ βa6 (with indices all different).
One then finds that the near-spacelike-singularity limit amounts to con-
sidering the large β limit in Eq.(2). In this limit a crucial role is played
by the linear forms wA(β) appearing in the “exponential walls”. Actually,
these walls enter in successive “layers”. A first layer consists of a sub-
set of all the walls called the dominant walls wi(β). The effect of these
dynamically dominant walls is to confine the motion in β-space to a fun-
damental billiard chamber defined by the inequalities wi(β) > 0. In the
case of SUGRA11, one finds that there are 10 dominant walls: 9 of them
are the symmetry walls wS12(β), w
23(β), ..., w
910(β), and the 10th is an elec-
tric 3-form wall e123(β) = β
1 + β2 + β3. As noticed in [6] a remarkable
fact is that the fundamental cosmological billiard chamber of SUGRA11
(as well as type-II string theories) is the Weyl chamber of the hyperbolic
Kac-Moody algebra E10. More precisely, the 10 dynamically dominant wall
forms
wS12(β), w
23(β), ..., w
910(β), e123(β)
can be identified with the 10 sim-
ple roots {α1(h), α2(h), ..., α10(h)} of E10. Here h parametrizes a generic el-
ement of a Cartan subalgebra (CSA) of E10 . [Let us also note that for
Heterotic and type-I string theories the cosmological billiard is the Weyl
310 dimensional for SUGRA11; but the various superstring theories also lead to a 10
dimensional Lorentz space because one must add the (positive) kinetic term of the dilaton
ϕ ≡ β10 to the 9-dimensional DeWitt metric corresponding to the 9 spatial dimensions.
chamber of another rank-10 hyperbolic Kac-Moody algebra, namely BE10].
In the Dynkin diagram of E10, Fig. 1, the 9 “horizontal” nodes correspond
to the 9 symmetry walls, while the characteristic “exceptional” node sticking
out “vertically” corresponds to the electric 3-form wall e123 = β
1 + β2 + β3.
[The fact that this node stems from the 3rd horizontal node is then seen to
be directly related to the presence of the 3-form Aµνλ, with electric kinetic
energy ∝ giℓgjmgknȦijkȦℓmn].
α1 α2 α3 α4 α5 α6 α7 α8 α9
✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐
Figure 1: Dynkin diagram of E10.
The appearance of E10 in the BKL behaviour of SUGRA11 revived an old
suggestion of B. Julia [12] about the possible role of E10 in a one-dimensional
reduction of SUGRA11. A posteriori, one can view the BKL behaviour as
a kind of spontaneous reduction to one dimension (time) of a multidimen-
sional theory. Note, however, that we are always discussing generic inho-
mogeneous 11-dimensional solutions, but that we examine them in the near-
spacelike-singularity limit where the spatial derivatives are sub-dominant:
∂x ≪ ∂t. Note also that the discrete E10(Z) was proposed as a U -duality
group of the full (T 10) spatial toroidal compactification of M-theory by Hull
and Townsend [13].
3 Gravity/Coset correspondence
Refs [8, 14] went beyond the leading-order BKL analysis just recalled by in-
cluding the first three “layers” of spatial-gradient-related sub-dominant walls
∝ exp(−2wA(β)) in Eq.(2). The relative importance of these sub-dominant
walls, which modify the leading billiard dynamics defined by the 10 dom-
inant walls wi(β), can be ordered by means of an expansion which counts
how many dominant wall forms wi(β) are contained in the exponents of the
sub-dominant wall forms wA(β), associated to higher spatial gradients. By
mapping the dominant gravity wall forms wi(β) onto the corresponding E10
simple roots αi(h), i = 1, ..., 10, the just described BKL-type gradient ex-
pansion becomes mapped onto a Lie-algebraic height expansion in the roots
of E10. It was remarkably found that, up to height 30 (i.e. up to small
corrections to the billiard dynamics associated to the product of 30 leading
walls e−2wi(β)), the SUGRA11 dynamics for gµν(t,x), Aµνλ(t,x) considered
at some given spatial point x0, could be identified to the geodesic dynam-
ics of a massless particle moving on the (infinite-dimensional) coset space
E10/K(E10). Note the “holographic” nature of this correspondence between
an 11-dimensional dynamics on one side, and a 1-dimensional one on the
other side.
A point on the coset space E10(R)/K(E10(R)) is coordinatized by a time-
dependent (but spatially independent) element of the E10(R) group of the
(Iwasawa) form: g(t) = exp h(t) exp ν(t). Here, h(t) = βacoset(t)Ha belongs
to the 10-dimensional CSA of E10, while ν(t) =
α>0 ν
α(t)Eα belongs to a
Borel subalgebra of E10 and has an infinite number of components labelled
by a positive root α of E10. The (null) geodesic action over the coset space
E10/K(E10) takes the simple form
SE10/K(E10) =
(vsym|vsym) (3)
where vsym ≡ 1
(v + vT ) is the “symmetric”4 part of the “velocity” v ≡
(dg/dt)g−1 of a group element g(t) running over E10(R).
The correspondence between the gravity, Eq. (2), and coset, Eq. (3), dy-
namics is best exhibited by decomposing (the Lie algebra of) E10 with respect
4Here the transpose operation T denotes the negative of the Chevalley involution ω
defining the real form E10(10) of E10. It is such that the elements k of the Lie sub-algebra
ofK(E10) are “T -antisymmetric”: k
T = −k, which is equivalent to them being fixed under
ω : ω(k) = +ω(k).
to (the Lie algebra of) the GL(10) subgroup defined by the horizontal line in
the Dynkin diagram of E10. This allows one to grade the various components
of g(t) by their GL(10) level ℓ. One finds that, at the ℓ = 0 level, g(t) is
parametrized by the Cartan coordinates βacoset(t) together with a unimodu-
lar upper triangular zehnbein θacoset i(t). At level ℓ = 1, one finds a 3-form
Acosetijk (t); at level ℓ = 2, a 6-form A
coset
i1i2...i6
(t), and at level ℓ = 3 a 9-index
object Acoseti1|i2...i9(t) with Young-tableau symmetry {8, 1}. The coset action
(3) then defines a coupled set of equations of motion for βacoset(t), θ
coset i(t),
Acosetijk (t), A
coset
i1...i6
(t), Acoseti1|i2...i9(t). By explicit calculations, it was found that
these coupled equations of motion could be identified (modulo terms corre-
sponding to potential walls of height at least 30) to the SUGRA11 equations
of motion, considered at some given spatial point x0.
The dictionary between the two dynamics says essentially that:
(0) βagravity(t,x0) ↔ βacoset(t) , θai (t,x0) ↔ θacoset i(t), (1) ∂t Acosetijk (t) corre-
sponds to the electric components of the 11-dimensional field strength Fgravity
= dAgravity in a certain frame e
i, (2) the conjugate momentum of Acoseti1...i6(t)
corresponds to the dual (using εi1i2...i10) of the “magnetic” frame compo-
nents of the 4-form Fgravity = dAgravity, and (3) the conjugate momentum of
Ai1|i2...i9(t) corresponds to the ε
10 dual (on jk) of the structure constants C ijk
of the coframe ei (d ei = 1
C ijk e
j ∧ ek).
The fact that at levels ℓ = 2 and ℓ = 3 the dictionary between supergrav-
ity and coset variables maps the first spatial gradients of the SUGRA variables
Aijk(t,x) and gij(t,x) onto (time derivatives of) coset variables suggested
the conjecture [8] of a hidden equivalence between the two models, i.e. the
existence of a dynamics-preserving map between the infinite tower of (spa-
tially independent) coset variables (βacoset, ν
α), together with their conjugate
momenta (πcoseta , pα), and the infinite sequence of spatial Taylor coefficients
(β(x0), π(x0), Q(x0), P (x0), ∂Q(x0), ∂
2β(x0), ∂
2Q(x0), . . . , ∂
nQ(x0), . . .)
formally describing the dynamics of the gravity variables (β(x), π(x), Q(x),
P (x)) around some given spatial point x0.
5One, however, expects the map between the two models to become spatially non-local
It has been possible to extend the correspondence between the two models
to the inclusion of fermionic terms on both sides [15, 16, 17]. Moreover,
Ref. [18] found evidence for a nice compatibility between some high-level
contributions (height −115!) in the coset action, corresponding to imaginary
roots6, and M-theory one-loop corrections to SUGRA11, notably the terms
quartic in the curvature tensor. (See also [19] for a study of the compatibility
of an underlying Kac-Moody symmetry with quantum corrections in various
models).
4 A new view of the (quantum) fate of space
at a cosmological singularity
Let us now, following [10], sketch the physical picture suggested by the
gravity/coset correspondence. That is, let us take seriously the idea that,
upon approaching a spacelike singularity, the description in terms of a spa-
tial continuum, and space-time based (quantum) field theory breaks down,
and should be replaced by a purely abstract Lie algebraic description. More
precisely, we suggest that the information previously encoded in the spatial
variation of the geometry and of the matter fields gets transferred to an
infinite tower of spatially independent (but time dependent) Lie algebraic
variables. In other words, we are led to the conclusion that space actually
“disappears” (or “de-emerges”) as the singularity is approached7. In partic-
ular (and this would be bad news for Gabriele’s pre-big bang scenario), we
suggest no (quantum) “bounce” from an incoming collapsing universe to some
outgoing expanding universe. Rather it is suggested that “life continues” for
for heights ≥ 30.
6i.e. such that (α, α) < 0, by contrast to the “real” roots, (α, α) = +2, which enter the
checks mentionned above.
7We have in mind here a “big crunch”, i.e. we conventionally consider that we are
tending towards the singularity. Mutatis mutandis, we would say that space “appears” or
“emerges” at a big bang.
an infinite “affine time” at a singularity, with the double understanding,
however, that: (i) life continues only in a totally new form (as in a kind of
“transmigration”), and (ii) an infinite affine time interval (measured, say, in
the coordinate t of Eq. (3) with a coset lapse function n(t) = 1) corresponds
to a sub-Planckian interval of geometrical proper time8.
Let us also comment on some expected aspects of the “duality” between
the two models. It seems probable (from the AdS/CFT paradigm) that,
even if the equivalence between the “gravity” and the “coset” descriptions
is formally exact, each model has a natural domain of applicability in which
the corresponding description is sufficiently “weakly coupled” to be trustable
as is, even in the leading approximation. For the gravity description this
domain is clearly that of curvatures smaller than the Planck scale. One then
expects that the natural domain of validity of the dual coset model would
correspond (in gravity variables) to that of curvatures larger than the Planck
scale. In addition, it is possible that the coset description should primarily
be considered as a quantum model, as now sketched.
The coset action (3) describes the classical motion of a massless particle
on the symmetric space E10(R)/K(E10(R)). Quantum mechanically, one
should consider a quantum massless particle, i.e., if we neglect polarization
effects9 a Klein-Gordon equation,
�Ψ(βa, να) = 0 , (4)
where � denotes the (formal) Laplace-Beltrami operator on the infinite-
dimensional Lorentz-signature curved coset manifold E10(R)/K(E10(R)).
Eq. (4) would apply to the case considered here of un-compactified M-theory.
In the case where all spatial dimensions are toroidally compactified, it has
been suggested [20, 21] that Ψ satisfy (4) together with a condition of period-
8Indeed, it is found that the coset time t (with n(t) = 1) corresponds to a “Zeno-like”
gravity coordinate time (with rescaled lapse Ñ = N/
g = 1) which tends to +∞ as the
proper time tends to zero.
9Actually, Refs. [15, 16, 17] indicate the need to consider a spinning massless particle,
i.e. some kind of Dirac equation on E10/K(E10).
icity over the discrete group E10(Z). In other words, Ψ would be a “modular
wave form” on E10(Z)\E10(R)/K(E10(R)).
Let us emphasize (still following [10]) that all reference to space and
time has disappeared in Eq. (4). The disappearance of time is common
between (4) and the usual Wheeler-DeWitt equation in which the “wave
function(al) of the universe” Ψ[gij(x)] no longer depends on any extrinsic time
parameter. [As usual, one needs to choose, among all the dynamical variables
a specific “clock field” to be used as an intrinsic time variable parametrizing
the dynamics of the remaining variables.] The interesting new feature of (4)
(when compared to a Wheeler-DeWitt type equation) is the disappearance
of any notion of geometry gij(x) and its replacement by the infinite tower
of Lie-algebraic variables (βa, να)10. This quantum de-emergence of space,
and the emergence of an infinite-dimensional symmetry group E10
11 which
deeply intertwines space-time with matter degrees of freedom might be radical
enough to get us closer to an understanding of the fate of space-time and
matter at cosmological singularities.
Acknowledgments. It is a pleasure to dedicate this review to Gabriele
Veneziano, a dear friend and a great physicist from whom I have learned a
lot. I am also very grateful to my collaborators Marc Henneaux and Hermann
Nicolai for the (continuing) E10 adventure. I wish also to thank Maurizio
Gasperini and Jnan Maherana for their patience.
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