As we have indicated above, the emergence of hyperbolic Coxeter groups in the BKL-limit has been argued to be the tip of an iceberg signaling the existence of a huge number of symmetries underlying gravitational theories. However, while the appearance of hyperbolic Coxeter groups is a solid fact that will in our opinion survive future developments, the exact way in which the conjectured infinite-dimensional symmetry acts is still a matter of debate and research.

The aim of this section is to describe one line of investigation for making the infinite-dimensional symmetry manifest. This approach is directly inspired by the results obtained through toroidal dimensional reduction of gravitational theories, where the scalar fields form coset manifolds exhibiting explicitly larger and larger symmetries as one goes down in dimensions. In the case of eleven-dimensional supergravity, reduction on an -torus reveals a chain of exceptional U-duality symmetries [33, 34], culminating with in three dimensions [134]. This has lead to the conjecture [113] that the chain of enhanced symmetries should in fact remain unbroken and give rise to the infinite-dimensional duality groups and , as one reduces the theory to two, one and zero dimensions, respectively.

The connection between the symmetry groups controlling the billiards in the BKL-limit, and the symmetry groups appearing in toroidal dimensional reduction to three dimensions, where coset spaces play a central role, has led to the attempt to reformulate eleven-dimensional supergravity as a -dimensional nonlinear sigma model based on the infinite-dimensional coset manifold [47]. This sigma model describes the geodesic flow of a particle moving on , whose dynamics can be seen to match the dynamics of the associated (suitably truncated) supergravity theory. Another, related, source of inspiration for the idea pushed forward in [47] has been the earlier proposal to reformulate eleven-dimensional supergravity as a nonlinear realisation of the even bigger symmetry [167], containing as a subgroup.

A central tool in the analysis of [47] is the level decomposition studied in Section 8. Although proposed some time ago and crowned with partial successes at low levels, the attempt to reformulate eleven-dimensional supergravity as an infinite-dimensional nonlinear sigma model, faces obstacles that have not yet been overcome at higher levels. This indicates that novel ideas are needed in order to make further progress towards a complete understanding of the role of infinite-dimensional symmetry groups in gravitational theories.

We begin by describing some general aspects of nonlinear sigma models for finite-dimensional coset spaces. We then explain how to generalize the construction to the infinite-dimensional case. We finally apply the construction in detail to the case of eleven-dimensional supergravity where the conjectured symmetry group is . This is one of the most extensively investigated models in the literature in view of its connection with M-theory. The techniques presented, however, can be applied to all gravitational models exhibiting the -duality symmetries discussed in Sections 5 and 7.

9.1 Nonlinear sigma models on finite-dimensional coset spaces

9.1.1 The Cartan involution and symmetric spaces

Example: The coset space

9.1.2 Nonlinear realisations

9.1.3 Three ways of writing the quadratic -invariant action

9.1.4 Equations of motion and conserved currents

9.1.5 Example: (hyperbolic space)

9.1.6 Parametrization of

9.2 Geodesic sigma models on infinite-dimensional coset spaces

9.2.1 Formal construction

9.2.2 Consistent truncations

Subgroup truncation

Level truncation and height truncation

9.3 Eleven-dimensional supergravity and

9.3.1 Low level fields

9.3.2 The -sigma model

9.3.3 Sigma model fields and -covariance

9.3.4 “Covariant derivatives”

9.3.5 The -invariant action at low levels

9.3.6 The correspondence

Levels 0 and 1

Level 2

Level 3

The dictionary

9.3.7 Higher levels and spatial gradients

The “gradient conjecture”

U-duality and the Weyl Group of

9.4 Further comments

9.4.1 Massive type IIA supergravity

9.4.2 Including fermions

9.4.3 Quantum corrections

9.4.4 Understanding duality

9.1.1 The Cartan involution and symmetric spaces

Example: The coset space

9.1.2 Nonlinear realisations

9.1.3 Three ways of writing the quadratic -invariant action

9.1.4 Equations of motion and conserved currents

9.1.5 Example: (hyperbolic space)

9.1.6 Parametrization of

9.2 Geodesic sigma models on infinite-dimensional coset spaces

9.2.1 Formal construction

9.2.2 Consistent truncations

Subgroup truncation

Level truncation and height truncation

9.3 Eleven-dimensional supergravity and

9.3.1 Low level fields

9.3.2 The -sigma model

9.3.3 Sigma model fields and -covariance

9.3.4 “Covariant derivatives”

9.3.5 The -invariant action at low levels

9.3.6 The correspondence

Levels 0 and 1

Level 2

Level 3

The dictionary

9.3.7 Higher levels and spatial gradients

The “gradient conjecture”

U-duality and the Weyl Group of

9.4 Further comments

9.4.1 Massive type IIA supergravity

9.4.2 Including fermions

9.4.3 Quantum corrections

9.4.4 Understanding duality

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