Chaos and Symmetry in String Cosmology
arXiv:0704.1457v1  [gr-qc]  11 Apr 2007
Chaos and Symmetry in String Cosmology1
Thibault Damour
Institut des Hautes Etudes Scientifiques, 35 route de Chartres,
F-91440 Bures-sur-Yvette, France
Abstract: We review the recently discovered interplay between
chaos and symmetry in the general inhomogeneous solution of many
string-related Einstein-matter systems in the vicinity of a cosmo-
logical singularity. The Belinsky-Khalatnikov-Lifshitz-type chaotic
behaviour is found, for many Einstein-matter models (notably those
related to the low-energy limit of superstring theory andM-theory),
to be connected with certain (infinite-dimensional) hyperbolic Kac-
Moody algebras. In particular, the billiard chambers describing
the asymptotic cosmological behaviour of pure Einstein gravity in
spacetime dimension d + 1, or the metric-three-form system of 11-
dimensional supergravity, are found to be identical to the Weyl
chambers of the Lorentzian Kac-Moody algebras AEd, or E10, re-
spectively. This suggests that these Kac-Moody algebras are hidden
symmetries of the corresponding models. There even exists some
evidence of a hidden equivalence between the general solution of
the Einstein-three-form system and a null geodesic in the infinite
dimensional coset space E10/K(E10), where K(E10) is the maximal
compact subgroup of E10.
1Invited talk at the 11th Marcel Grossmann Meeting on Recent Developments in Gen-
eral Relativity, Berlin, Germany, 23-29 July 2006.
http://arxiv.org/abs/0704.1457v1
1 Introduction
We wish to review a recently discovered intriguing connection between two,
seemingly antagonistic, structures present in pure Einstein gravity, and in
some, string-theory motivated, Einstein-matter systems.
On the one hand, Belinsky, Khalatnikov and Lifshitz (BKL) [1] discov-
ered that the asymptotic behaviour of the general solution of the (3 + 1)-
dimensional Einstein’s equations, in the vicinity of a cosmological singularity,
exhibited a chaotic structure: they showed that, because of non-linearities in
Einstein equations, the generic, inhomogeneous solution behaves as a chaotic
[2] sequence of “generalized Kasner solutions”. They then showed that pure
gravity in 4+1 dimensions exhibits a similar chaotic structure [3]. The exten-
sion of the BKL analysis to pure gravity in higher dimensions was addressed
in [4, 5]. A surprising result was found: while the general behaviour of the
vacuum Einstein solutions remains “chaotic” (i.e. “oscillatory”) for space-
time dimensions D ≤ 10, it ceases to be so for spacetime dimensions D ≥ 11,
where it becomes monotonic and Kasner-like.
On the other hand, a symmetry structure was found to be present (often
in a hidden form) in the “dimensional reductions” of Einstein’s gravity. The
paradigm of this hidden symmetry structure is the continuous SL(2,R)E
symmetry group found long ago by Ehlers [6] for D = 4 Einstein gravity in
the presence of one Killing vector. When two commuting Killing vectors are
present, the finite-dimensional Ehlers group SL(2,R) was found, through,
notably, the work of Matzner and Misner, of Geroch, of Julia, of Breiten-
lohner and Maison, and of Belinsky and Zakharov, to be promoted to an
infinite-dimensional symmetry group, which can be identified to the affine
Kac-Moody extension of SL(2,R) (see [7] for references and a review). A
similar pattern was found to take place in supergravity theories, and notably
in the “maximal” supergravity theory which lives in D = 11. In that case, it
was remarkably found [8] that the successive toroidal dimensional reductions
of D = 11 supergravity to lower dimensions (in the presence of an increasing
number of commuting Killing vectors) admitted a correspondingly larger and
larger hidden symmetry group En(R), where n denotes the number of Killing
vectors, and En the (extended) sequence of the groups belonging to the excep-
tional series in the Cartan-Killing classification of finite-dimensional simple
Lie groups. The latter series culminates into the last finite-dimensional ex-
ceptional group E8 in the case where one has 8 Killing vectors, i.e. in the
case of toroidal compactification (on T 8) down to a D′ = 3 dimensionally
reduced theory. However, as first conceived by Julia [9], this symmetry-
increasing pattern is expected to continue to the (infinite-dimensional) affine
Kac-Moody group E9 for the reduction of supergravity down to D
′ = 2 (9
Killing vectors). This was indeed explicitly proven later [10]. It was also
mentionned by Julia [9] that the still larger symmetry group E10 (which is
now an infinite-dimensional hyperbolic Kac-Moody group2) might arise when
trying to further reduce maximal supergravity. However, this poses a serious
challenge because the naive dimensional reduction to D′ = 1, i.e. the set of
D = 11 supergravity solutions which depend only on one (time) variable is
much too small to (faithfully) carry such a huge symmetry.
At this point, the two separate threads (chaos and symmetry) in our
history unexpectedly merged in a sequence of works which, while extend-
ing the BKL analysis to D = 11 supergravity, and to the D = 10 models
describing the low-energy limit of the various superstring theories [12, 13],
found that behind their seeming entirely chaotic3 BKL-type behaviour there
were hints of a hidden symmetry structure [14, 15, 16] linked to E10. More
precisely, it was first found [14] that the “billiard chamber” (see [17] and
below) describing the BKL-type behaviour of D = 11 supergravity (as well
as of D = 10 IIA and IIB superstring theories) could be identified with the
“Weyl chamber” of E10. [The cosmological billiard chambers of type I and
heterotic string theories coincides with the Weyl chamber of BE10. We fo-
cus here on the more fundamental E10 case.] Similarly, an examination of
2See [11] for an introduction to the theory of Kac-Moody algebras.
3Though D = 11 pure gravity is monotonically Kasner-like instead of chaotic, the
presence of the three-form A in D = 11 supergravity was found to reintroduce a generic
chaotic, oscillatory behaviour.
the case of pure gravity in spacetime dimension D ≡ d + 1 revealed the
presence of the Weyl chamber of the Lorentzian Kac-Moody algebras AEd
(which are hyperbolic only when d ≤ 9) [15]. Then it was shown that, up
to height 30 in a “height expansion” related to the BKL “gradient expan-
sion”, the dynamics of the 11-dimensional supergravity variables could be
identified with the dynamics of a null geodesic on the infinite-dimensional
coset space E10/K(E10), where K(E10) is the maximal compact subgroup of
E10 [16]. This led to the conjecture that there exists a hidden equivalence
between D = 11 supergravity and null geodesic motion on E10/K(E10). This
conjecture can be generalized to other gravity models and other coset spaces
[17]. In particular, usual (3 + 1)-dimensional gravity might be equivalent to
null geodesic motion on AE3/K(AE3). If these conjectures were true, they
would mean that the infinite-dimensional numerator Kac-Moody groups, E10
or AE3, are hidden continuous “symmetries” of the respective field equations
(supergravity11 or gravity4) which transform solutions into (new) solutions.
In the case of D = 4 gravity, this would represent a huge generalization4 of
the Ehlers SL(2,R) symmetry group.
The present paper is organized as follows. In Section 2 we summa-
rize the Hamiltonian “cosmological billiard” approach to BKL behaviour
(mainly drawing from [17]). See [18] and the contribution of C. Uggla to
these proceedings for a comparison between this Hamiltonian cosmological
billiard approach and a “Hubble-normalized” “dynamical systems picture”.
In Section 3 we sketch the “correspondence” between a gravity model (e.g.
D = 11 supergravity) and a coset geodesic dynamics (e.g. geodesic mo-
tion on E10/K(E10)). For more details on the construction and structure of
those Kac-Moody coset models see the contribution of A. Kleinschmidt and
H. Nicolai to these proceedings. Finally, Section 4 offers some (speculative)
conclusions.
4This is a generalization in two separate senses: (i) the conjectured AE3 symmetry
would apply in absence of any Killing field, and (ii) AE3 is the hyperbolic Kac-Moody
group canonically associated with A1 = SL(2) (while the “Geroch group” is the affine
Kac-Moody group associated with A1 = SL(2)).
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 [1]. To see this, it is convenient [17] 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-dimensional5 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 [14] 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
510 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 the
old suggestion of [9] about the possible role of E10 in a one-dimensional re-
duction of SUGRA11. A posteriori, one can view the BKL behaviour as a
kind of spontaneous reduction to one dimension (time) of a multidimensional
theory. Note, however, that we are always discussing generic inhomogeneous
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 [19].
3 Gravity/Coset correspondence
Refs [16, 20] 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”6 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
6Here 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 [16] 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.
7One, 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 [21, 22, 23]. Moreover,
Ref. [24] found evidence for a nice compatibility between some high-level
contributions (height −115!) in the coset action, corresponding to imaginary
roots8, and M-theory one-loop corrections to SUGRA11, notably the terms
quartic in the curvature tensor. (See also [25] for a study of the compatibility
of an underlying Kac-Moody symmetry with quantum corrections in various
models).
4 Conclusions and outlook
At this stage, we are far from having a proof of the full equivalence between
SUGRA11 and the E10/K(E10) model, or of any other gravity/coset con-
jectured pair. The partial evidence summarized above is suggestive of the
existence of some kind of hidden symmetry structure in General Relativity
and Supergravity. However, it is quite possible that this hidden symmetry
is present only in a way which cannot be explicitly realized at the level of
the classical field equations. Indeed, the situation might be similar to that
discussed in the plenary talks of Sasha Polyakov, Igor Klebanov and Eva
Silverstein. In the much better understood gravity/gauge correspondence
a quasi-classical, weakly curved spacetime corresponds to a strongly quan-
tum, strongly self-interacting gauge theory state. Reciprocally, the perturba-
tive, weakly-interacting gauge theory states correspond to non-perturbative,
strongly curved gravity states. By analogy, we might expect that the sim-
ple geodesic coset motions correspond to strongly curved spacetimes, with
curvatures larger than the Planck scale. [This would intuitively explain why
the coset picture becomes prominent in the formal limit where one tends
towards an infinite-curvature singularity.] In addition, it is possible that the
for heights ≥ 30.
8i.e. such that (α, α) < 0, by contrast to the “real” roots, (α, α) = +2, which enter the
checks mentionned above.
equivalence between the gravity and coset models hold only at the level of
the quantized models. In this respect, note that the quantum version of the
null geodesic dynamics (3) would be, if we neglect polarization effects9 a
Klein-Gordon equation10,
�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)).
As recently emphasized [28], the gravity/coset correspondence sketched
above suggests a new physical picture of the fate of space at a cosmological
singularity. It suggests that, upon approaching a spacelike singularity, the
description in terms of a spatial continuum breaks down and should be re-
placed by a purely abstract Lie algebraic description. In other words, one is
led to the conclusion that space actually “disappears” (or “de-emerges”) as
the singularity is approached11. The gravity/coset duality suggests that there
is no (quantum) “bounce” from an incoming collapsing universe to some out-
going expanding one [24]. Rather, it is suggested that “life continues” for an
infinite “affine time” at a singularity, with the double understanding 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 time12.
9Actually, Refs [21, 22, 23] indicate the need to consider a spinning massless particle,
i.e. some kind of Dirac equation on E10/K(E10).
10This equation, submitted to a condition of periodicity over a discrete group E10(Z),
has been considered in Refs [26, 27] in the context of quantum theories with all spatial
dimensions being toroidally compactified.
11We have in mind here a “big crunch”, i.e. we conventionally consider that we are
tending toward the singularity. Mutatis mutandis, we would say that space “appears” or
“emerges” at a big bang.
12Indeed, 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.
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