Spacelike Singularities and Hidden Symmetries of Gravity

Abstract

We review the intimate connection between (super-)gravity close to a spacelike singularity (the “BKL-limit”) and the theory of Lorentzian Kac-Moody algebras. We show that in this limit the gravitational theory can be reformulated in terms of billiard motion in a region of hyperbolic space, revealing that the dynamics is completely determined by a (possibly infinite) sequence of reflections, which are elements of a Lorentzian Coxeter group. Such Coxeter groups are the Weyl groups of infinite-dimensional Kac-Moody algebras, suggesting that these algebras yield symmetries of gravitational theories. Our presentation is aimed to be a self-contained and comprehensive treatment of the subject, with all the relevant mathematical background material introduced and explained in detail. We also review attempts at making the infinite-dimensional symmetries manifest, through the construction of a geodesic sigma model based on a Lorentzian Kac-Moody algebra. An explicit example is provided for the case of the hyperbolic algebra E10, which is conjectured to be an underlying symmetry of M-theory. Illustrations of this conjecture are also discussed in the context of cosmological solutions to eleven-dimensional supergravity.

On theE10/massive type IIA supergravity correspondence

“…This analysis mainly focused on the bosonic sector, but a preliminary analysis of the fermionic sector was initiated by restricting to the zeroth level of the e 10 -decomposition. It was also pointed out in [14,37,54] that the mass term has a natural interpretation as a generator at level four in a decomposition of e 10 with respect to A 8 ∼ = sl(9, R). From this point of view, the kinetic term in the sigma model associated with this generator naturally gives rise to the mass term of type IIA upon dimensional reduction.…”

“…In order to incorporate also the fermionic sector in the sigma model, we will analyse the relevant (unfaithful) Dirac-spinor and vector-spinor representations of k(e 10 ) up to the desired level. 5 For a more detailed discussion of the general e 10 methods employed here we refer to the review papers [52][53][54], for more information about k(e 10 ) see [17][18][19]21], and for a mathematical introduction to Kac-Moody algebras the canonical reference is [55].…”

Gravitation and quadratic forms

“…Simple mathematical structures are often signatures of a hidden symmetry and this makes them interesting. There is growing evidence that pure gravity in d = 4 may have hidden symmetries [2] and that some of the surprising ultraviolet cancelations encountered in the N = 8 theory may originate from pure gravity itself [3] -unexpected cancelations themselves being another reliable indicator of hidden symmetries. We hope that the quadratic form structures introduced in this paper are signatures of such a symmetry, with possible links to the recent work of [4].…”

Kac-Moody and Borcherds symmetries of six-dimensional chiral supergravity

“…oscillatory behaviour near a spacelike singularity of gravitational models [18][19][20][21] as a billiard motion in the Weyl chamber of a hyperbolic Kac-Moody algebra, paving the way to a different non-linear realization of the symmetry where space and time are on distinct footings [22] (for reviews, see [23,24]). 1 Non-linear realizations of the infinite-dimensional conjectured symmetry spectacularly encode the correct field content of the corresponding supergravity theories, as well as the correct Chern-Simons couplings.…”