Gradient Representations and Affine Structures in AE(n)

TL;DR

This paper explores the structure of AE(n) Kac-Moody algebras in the context of dimensional reduction in Einstein's theory, revealing new algebraic relations and mappings to sigma-models and geodesic trajectories.

Contribution

It introduces the commutation relations of gradient generators in AE(3) and connects truncated sigma-models to Einstein's equations and geodesic trajectories on infinite-dimensional cosets.

Findings

Derived commutation relations for gradient generators in AE(3).

Mapped low level AE(n) sigma-model truncations to Einstein's reduced equations.

Established a correspondence between diagonal solutions and null geodesics on coset spaces.

Abstract

We study the indefinite Kac-Moody algebras AE(n), arising in the reduction of Einstein's theory from (n+1) space-time dimensions to one (time) dimension, and their distinguished maximal regular subalgebras sl(n) and affine A_{n-2}^{(1)}. The interplay between these two subalgebras is used, for n=3, to determine the commutation relations of the `gradient generators' within AE(3). The low level truncation of the geodesic sigma-model over the coset space AE(n)/K(AE(n)) is shown to map to a suitably truncated version of the SL(n)/SO(n) non-linear sigma-model resulting from the reduction Einstein's equations in (n+1) dimensions to (1+1) dimensions. A further truncation to diagonal solutions can be exploited to define a one-to-one correspondence between such solutions, and null geodesic trajectories on the infinite-dimensional coset space H/K(H), where H is the (extended) Heisenberg group, and K(H) its maximal compact subgroup. We clarify the relation between H and the corresponding subgroup of the Geroch group.