The Geroch Group in One Dimension
Abstract
We study the dimensional reduction of general relativity to a single null spacetime dimension. The dimensionally reduced theory is a theory of six scalar fields governed by three constraints. It has an infinite dimensional symmetry which is an enhanced version of the Geroch group. To get a local action of the symmetry on solution space, we need to introduce an infinite tower of new fields and new constraints. The symmetry appears to be a hyperbolic KacMoody algebra, with the caveat that some of the defining relations of the hyperbolic KacMoody algebra are only checked ``order by order'' on the infinite tower of new fields. This is a very mysterious Lie algebra with no known geometrical interpretation. It is not even clear how to enumerate a basis. We explore this problem using the action of the algebra on solution space and find an intriguing connection to the representation theory of the symmetric group. The symmetry described here might be related to the dynamics of gravity near spacelike singularities.
 Publication:

arXiv eprints
 Pub Date:
 December 2021
 arXiv:
 arXiv:2112.05661
 Bibcode:
 2021arXiv211205661P
 Keywords:

 High Energy Physics  Theory;
 General Relativity and Quantum Cosmology;
 Mathematics  Representation Theory
 EPrint:
 20 pages