Cosmological Singularities and a Conjectured Gravity/Coset Correspondence

TL;DR

This paper explores a potential deep connection between the chaotic behavior of cosmological singularities in supergravity and the mathematical structure of hyperbolic Kac-Moody algebras, proposing a new perspective on the quantum nature of space at singularities.

Contribution

It reviews the possible correspondence between supergravity near singularities and null geodesic motion on an infinite-dimensional coset space, suggesting a novel approach to understanding quantum cosmology.

Findings

Identifies a link between chaotic singularities and hyperbolic Kac-Moody algebra E(10)

Proposes a gravity/coset correspondence as a new theoretical framework

Suggests implications for the quantum fate of space at singularities

Abstract

We review the recently discovered connection between the Belinsky-Khalatnikov-Lifshitz-like ``chaotic'' structure of generic cosmological singularities in eleven-dimensional supergravity and the ``last'' hyperbolic Kac-Moody algebra E(10). This intriguing connection suggests the existence of a hidden ``correspondence'' between supergravity (or even M-theory) and null geodesic motion on the infinite-dimensional coset space K(E(10)). If true, this gravity/coset correspondence would offer a new view of the (quantum) fate of space (and matter) at cosmological singularities.

Equations

$$d{s}^2=-N^{2}d{t}^2+\sum _{a=1}^d{e}^{-2{\beta }^a}{\theta }_i^a{\theta }_j^{a}d{x}^{i}d{x}^j.$$

$$\mathcal{H}({\beta }^a,{\pi }_a,P,Q)$$

$$S_{E_{10}/K(E_{10})}=\int \frac{dt}{n(t)}(v^{\text{sym}}|v^{\text{sym}})$$

$$\square \Psi ({\beta }^a,{\nu }^{\alpha })=0,$$

Full text

Cosmological Singularities and a Conjectured Gravity/Coset Correspondence111Contribution to “String Theory and Fundamental Interactions” – in celebration of Gabriele Veneziano’s 65th birthday – eds. M. Gasperini and J. Maharana, Springer-Verlag, Heidelberg, 2007.

Thibault Damour

Institut des Hautes Etudes Scientifiques, 35 route de Chartres,

F-91440 Bures-sur-Yvette, France

Abstract: We review the recently discovered connection between the Belinsky-Khalatnikov-Lifshitz-like “chaotic” structure of generic cosmological singularities in eleven-dimensional supergravity and the “last” hyperbolic Kac-Moody algebra ${E_{10}}$. This intriguing connection suggests the existence of a hidden “correspondence” between supergravity (or even $M$-theory) and null geodesic motion on the infinite-dimensional coset space $E_{10}/K(E_{10})$. If true, this gravity/coset correspondence would offer a new view of the (quantum) fate of space (and matter) at cosmological singularities.

1 Introduction

It is a pleasure to participate in the celebration of the seminal accomplishments 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 Cosmology: 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 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 effective 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 seemingly very different dynamical systems: on the one hand, $11$-dimensional supergravity (or even, hopefully, “$M$-theory”), and, on the other hand, a one-dimensional $E_{10}/K(E_{10})$ nonlinear $\sigma$ model, i.e. the geodesic motion of a massless particle on the infinite-dimensional coset space222Here $K(E_{10})$ denotes the (formal) “maximal compact subgroup” of the hyperbolic Kac-Moody group $E_{10}$. $E_{10}/K(E_{10})$ [8]. The intuitive hope behind this conjecture is that the BKL-type near spacelike 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 confirmed, 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).

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 $g_{\mu\nu}(t,\textbf{x})$, together with the other fields (such as the 3-form $A_{\mu\nu\lambda}(t,\textbf{x})$ in supergravity), near a singular hypersurface. The basic idea is that, near a spacelike singularity, 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 $g_{\mu\nu}$ into non-dynamical (lapse $N$, and shift $N^{i}$, here set to zero) and dynamical ($e^{-2\beta^{a}}$, $\theta^{a}_{i})$ components. They are defined so that the line element reads

[TABLE]

Here $d\equiv D-1$ denotes the spatial dimension ($d=10$ for SUGRA11, and $d=9$ for string theory), $e^{-2\beta^{a}}$ represent (in an Iwasawa decomposition) the “diagonal” components of the spatial metric $g_{ij}$, while the “off diagonal” components are represented by the $\theta^{a}_{i}$, defined to be upper triangular matrices with 1’s on the diagonal (so that, in particular, $\det\theta=1$).

The Hamiltonian constraint, at a given spatial point, reads (with $\tilde{N}\equiv N/\sqrt{\det g_{ij}}$ denoting the “rescaled lapse”)

[TABLE]

Here $\pi_{a}$ (with $a=1,...,d$) denote the canonical momenta conjugate to the “logarithmic scale factors” $\beta^{a}$, while $Q$ denote the remaining configuration variables ($\theta^{a}_{i}$, 3-form components $A_{ijk}(t,\textbf{x})$ in supergravity), and $P$ their canonically conjugate momenta ($P^{i}_{a},\pi^{ijk}$). The symbol $\partial$ denotes spatial derivatives. The (inverse) metric $G^{ab}$ in Eq. (2) is the DeWitt “superspace” metric induced on the $\beta$’$s$ by the Einstein-Hilbert action. It endows the $d$-dimensional33310 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 $\varphi\equiv\beta^{10}$ to the 9-dimensional DeWitt metric corresponding to the 9 spatial dimensions. $\beta$ space with a Lorentzian structure $G_{ab}\,\dot{\beta}^{a}\dot{\beta}^{b}$.

One of the crucial features of Eq. (2) is the appearance of Toda-like exponential potential terms $\propto\exp(-2w_{A}(\beta))$, where the $w_{A}(\beta)$ are linear forms in the logarithmic scale factors: $w_{A}(\beta)\equiv w_{Aa}\,\beta^{a}$. The range of labels $A$ and the specific “wall forms” $w_{A}(\beta)$ that appear depend on the considered model. For instance, in SUGRA11 there appear: “symmetry wall forms” $w_{ab}^{S}(\beta)\equiv\beta^{b}-\beta^{a}$ (with $a<b$), “gravitational wall forms” $w_{abc}^{g}(\beta)\equiv 2\beta^{a}+\underset{e\neq a,b,c}{\sum}\,\beta^{e}$ ($a\neq b$, $b\neq c$, $c\neq a$), “electric 3-form wall forms”, $e_{abc}(\beta)\equiv\beta^{a}+\beta^{b}+\beta^{c}$ ($a\neq b$, $b\neq c$, $c\neq a$), and “magnetic 3-form wall forms”, $m_{a_{1}....a_{6}}\equiv\beta^{a_{1}}+\beta^{a_{2}}+...+\beta^{a_{6}}$ (with indices all different).

One then finds that the near-spacelike-singularity limit amounts to considering the large $\beta$ limit in Eq.(2). In this limit a crucial role is played by the linear forms $w_{A}(\beta)$ appearing in the “exponential walls”. Actually, these walls enter in successive “layers”. A first layer consists of a subset of all the walls called the dominant walls $w_{i}(\beta)$. The effect of these dynamically dominant walls is to confine the motion in $\beta$-space to a fundamental billiard chamber defined by the inequalities $w_{i}(\beta)\geqslant 0$. In the case of SUGRA11, one finds that there are 10 dominant walls: 9 of them are the symmetry walls $w_{12}^{S}(\beta),w_{23}^{S}(\beta),...,w_{910}^{S}(\beta),$ and the 10th is an electric 3-form wall $e_{123}(\beta)=\beta^{1}+\beta^{2}+\beta^{3}$. 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 $E_{10}$. More precisely, the 10 dynamically dominant wall forms \big{\{}w_{12}^{S}(\beta),w_{23}^{S}(\beta),...,w_{910}^{S}(\beta),e_{123}(\beta)\big{\}} can be identified with the 10 simple roots $\{\alpha_{1}(h),\alpha_{2}(h),...,\alpha_{10}(h)\}$ of $E_{10}$. Here $h$ parametrizes a generic element of a Cartan subalgebra (CSA) of $E_{10}$ . [Let us also note that for Heterotic and type-I string theories the cosmological billiard is the Weyl chamber of another rank-10 hyperbolic Kac-Moody algebra, namely $BE_{10}$]. In the Dynkin diagram of $E_{10}$, 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 $e_{123}=\beta^{1}+\beta^{2}+\beta^{3}$. [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 $A_{\mu\nu\lambda}$, with electric kinetic energy $\propto g^{i\ell}g^{jm}g^{kn}\dot{A}_{ijk}\dot{A}_{\ell mn}$].

The appearance of $E_{10}$ in the BKL behaviour of SUGRA11 revived an old suggestion of B. Julia [12] about the possible role of $E_{10}$ 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 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: $\partial_{x}\ll\partial_{t}$. Note also that the discrete $E_{10}(\mathbb{Z})$ 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 including the first three “layers” of spatial-gradient-related sub-dominant walls $\propto\exp(-2w_{A}(\beta))$ in Eq.(2). The relative importance of these sub-dominant walls, which modify the leading billiard dynamics defined by the 10 dominant walls $w_{i}(\beta)$, can be ordered by means of an expansion which counts how many dominant wall forms $w_{i}(\beta)$ are contained in the exponents of the sub-dominant wall forms $w_{A}(\beta)$, associated to higher spatial gradients. By mapping the dominant gravity wall forms $w_{i}(\beta)$ onto the corresponding $E_{10}$ simple roots ${\alpha}_{i}(h),i=1,...,10,$ the just described BKL-type gradient expansion becomes mapped onto a Lie-algebraic height expansion in the roots of $E_{10}$. 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 $e^{-2w_{i}(\beta)}$), the SUGRA11 dynamics for $g_{\mu\nu}(t,\textbf{x})$, $A_{\mu\nu\lambda}(t,\textbf{x})$ considered at some given spatial point $\textbf{x}_{0}$, could be identified to the geodesic dynamics of a massless particle moving on the (infinite-dimensional) coset space $E_{10}/K(E_{10})$. 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 $E_{10}(\mathbb{R})/K(E_{10}(\mathbb{R}))$ is coordinatized by a time-dependent (but spatially independent) element of the $E_{10}(\mathbb{R})$ group of the (Iwasawa) form: $g(t)=\exp h(t)\exp\nu(t)$. Here, $h(t)=\beta_{\textrm{coset}}^{a}(t)H_{a}$ belongs to the 10-dimensional CSA of $E_{10}$, while $\nu(t)=\sum_{\alpha>0}\nu^{\alpha}(t)E_{\alpha}$ belongs to a Borel subalgebra of $E_{10}$ and has an infinite number of components labelled by a positive root $\alpha$ of $E_{10}$. The (null) geodesic action over the coset space $E_{10}/K(E_{10})$ takes the simple form

[TABLE]

where $v^{\textrm{sym}}\equiv\frac{1}{2}(v+v^{T})$ is the “symmetric”444Here the transpose operation $T$ denotes the negative of the Chevalley involution $\omega$ defining the real form $E_{10(10)}$ of $E_{10}$. It is such that the elements $k$ of the Lie sub-algebra of $K(E_{10})$ are “$T$-antisymmetric”: $k^{T}=-k$, which is equivalent to them being fixed under $\omega:\omega(k)=+\,\omega(k)$. part of the “velocity” $v\equiv(dg/dt)g^{-1}$ of a group element $g(t)$ running over $E_{10}(\mathbb{R})$.

The correspondence between the gravity, Eq. (2), and coset, Eq. (3), dynamics is best exhibited by decomposing (the Lie algebra of) $E_{10}$ with respect to (the Lie algebra of) the $GL(10)$ subgroup defined by the horizontal line in the Dynkin diagram of $E_{10}$. This allows one to grade the various components of $g(t)$ by their $GL(10)$ level $\ell$. One finds that, at the $\ell=0$ level, $g(t)$ is parametrized by the Cartan coordinates $\beta_{\rm coset}^{a}(t)$ together with a unimodular upper triangular zehnbein $\theta_{{\rm coset}\,i}^{a}(t)$. At level $\ell=1$, one finds a 3-form $A_{ijk}^{\rm coset}(t)$; at level $\ell=2$, a 6-form $A_{i_{1}i_{2}\ldots i_{6}}^{\rm coset}(t)$, and at level $\ell=3$ a $9$-index object $A_{i_{1}\mid i_{2}\ldots i_{9}}^{\rm coset}(t)$ with Young-tableau symmetry $\{8,1\}$. The coset action (3) then defines a coupled set of equations of motion for $\beta_{\rm coset}^{a}(t)$, $\theta_{{\rm coset}\,i}^{a}(t)$, $A_{ijk}^{\rm coset}(t)$, $A_{i_{1}\ldots i_{6}}^{\rm coset}(t)$, $A_{i_{1}\mid i_{2}\ldots i_{9}}^{\rm coset}(t)$. By explicit calculations, it was found that these coupled equations of motion could be identified (modulo terms corresponding to potential walls of height at least 30) to the SUGRA11 equations of motion, considered at some given spatial point $\textbf{x}_{0}$.

The dictionary between the two dynamics says essentially that:

(0) $\beta_{\rm gravity}^{a}(t,\textbf{x}_{0})\leftrightarrow\beta_{\rm coset}^{a}(t)\,,\ \theta_{i}^{a}(t,\textbf{x}_{0})\leftrightarrow\theta_{{\rm coset}\,i}^{a}(t)$, (1) $\partial_{t}\,A_{ijk}^{\rm coset}(t)$ corresponds to the electric components of the 11-dimensional field strength $F_{\rm gravity}$ $=d\,A_{\rm gravity}$ in a certain frame $e^{i}$, (2) the conjugate momentum of $A_{i_{1}\ldots i_{6}}^{\rm coset}(t)$ corresponds to the dual (using $\varepsilon^{i_{1}i_{2}\ldots i_{10}}$) of the “magnetic” frame components of the 4-form $F_{\rm gravity}=d\,A_{\rm gravity}$, and (3) the conjugate momentum of $A_{i_{1}\mid i_{2}\ldots i_{9}}(t)$ corresponds to the $\varepsilon^{10}$ dual (on $jk$) of the structure constants $C_{jk}^{i}$ of the coframe $e^{i}$ ($d\,e^{i}=\frac{1}{2}\,C_{jk}^{i}\,e^{j}\wedge e^{k}$).

The fact that at levels $\ell=2$ and $\ell=3$ the dictionary between supergravity and coset variables maps the first spatial gradients of the SUGRA variables $A_{ijk}(t,\textbf{x})$ and $g_{ij}(t,\textbf{x})$ 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 (spatially independent) coset variables $(\beta_{\rm coset}^{a},\nu^{\alpha})$, together with their conjugate momenta $(\pi_{a}^{\rm coset},p_{\alpha})$, and the infinite sequence of spatial Taylor coefficients $(\beta(\textbf{x}_{0})$, $\pi(\textbf{x}_{0})$, $Q(\textbf{x}_{0})$, $P(\textbf{x}_{0})$, $\partial Q(\textbf{x}_{0})$, $\partial^{2}\beta(\textbf{x}_{0})$, $\partial^{2}Q(\textbf{x}_{0}),\ldots,\partial^{n}Q(\textbf{x}_{0}),\ldots)$ formally describing the dynamics of the gravity variables $(\beta(\textbf{x}),\pi(\textbf{x}),Q(\textbf{x})$, $P(\textbf{x}))$ around some given spatial point $\textbf{x}_{0}$.555One, however, expects the map between the two models to become spatially non-local for heights $\geq 30$.

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 roots666i.e. such that $(\alpha,\alpha)<0$, by contrast to the “real” roots, $(\alpha,\alpha)=+2$, which enter the checks mentionned above., 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 spatial 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 approached777We 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.. In particular (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 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 time888Indeed, it is found that the coset time $t$ (with $n(t)=1$) corresponds to a “Zeno-like” gravity coordinate time (with rescaled lapse $\tilde{N}=N/\sqrt{g}=1$) which tends to $+\infty$ as the proper time tends to zero..

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 $E_{10}({\mathbb{R}})/K(E_{10}({\mathbb{R}}))$. Quantum mechanically, one should consider a quantum massless particle, i.e., if we neglect polarization effects999Actually, Refs. [15, 16, 17] indicate the need to consider a spinning massless particle, i.e. some kind of Dirac equation on $E_{10}/K(E_{10})$. a Klein-Gordon equation,

[TABLE]

where $\square$ denotes the (formal) Laplace-Beltrami operator on the infinite-dimensional Lorentz-signature curved coset manifold $E_{10}({\mathbb{R}})/K(E_{10}({\mathbb{R}}))$. 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 $\Psi$ satisfy (4) together with a condition of periodicity over the discrete group $E_{10}({\mathbb{Z}})$. In other words, $\Psi$ would be a “modular wave form” on $E_{10}({\mathbb{Z}})\backslash E_{10}({\mathbb{R}})/K(E_{10}({\mathbb{R}}))$.

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” $\Psi[g_{ij}(\textbf{x})]$ 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 $g_{ij}(\textbf{x})$ and its replacement by the infinite tower of Lie-algebraic variables $(\beta^{a},\nu^{\alpha})$101010Note that this is conceptually very different from the $E_{11}$-based proposal of [22].. This quantum de-emergence of space, and the emergence of an infinite-dimensional symmetry group $E_{10}$111111Let us note that $E_{10}$ enjoys a similarly distinguished status among the (infinite-dimensional) hyperbolic Kac-Moody Lie groups as $E_{8}$ does in the Cartan-Killing classification of the finite-dimensional simple Lie groups [23]. 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) $E_{10}$ adventure. I wish also to thank Maurizio Gasperini and Jnan Maherana for their patience.