The Geroch Group in One Dimension
Robert F. Penna

TL;DR
This paper investigates the dimensional reduction of general relativity to one null dimension, revealing an infinite-dimensional symmetry related to hyperbolic Kac-Moody algebras, with potential implications for understanding gravity near singularities.
Contribution
It introduces an extended symmetry structure in a reduced gravity model, connecting it to hyperbolic Kac-Moody algebras and exploring its algebraic and geometric properties.
Findings
Identified an enhanced Geroch group as a hyperbolic Kac-Moody algebra.
Connected the algebra's action to symmetric group representation theory.
Suggested a link between the symmetry and gravity near spacelike singularities.
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 Kac-Moody algebra, with the caveat that some of the defining relations of the hyperbolic Kac-Moody 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…
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Taxonomy
TopicsBlack Holes and Theoretical Physics · Advanced Topics in Algebra · Nonlinear Waves and Solitons
