Twisted Self-Duality of Dimensionally Reduced Gravity and Vertex Operators

TL;DR

This paper introduces a new solution generating group for dimensionally reduced gravity, explores its connection with the Geroch group, and demonstrates how vertex operators can be used for algebraic solutions of Einstein's equations.

Contribution

It presents the dressing group as a novel solution generating method and links it with vertex operators for algebraic solution construction.

Findings

The dressing group acts transitively on a dense subset of moduli space.

A new Lax pair reveals a twisted self-duality in the system.

Vertex operators enable algebraic solutions to Einstein's equations.

Abstract

The Geroch group, isomorphic to the SL(2,R) affine Kac-Moody group, is an infinite dimensional solution generating group of Einstein's equations with two surface orthogonal commuting Killing vectors. We introduce another solution generating group for these equations, the dressing group, and discuss its connection with the Geroch group. We show that it acts transitively on a dense subset of moduli space. We use a new Lax pair expressing a twisted self-duality of this system and we study the dressing problem associated to it. We also describe how to use vertex operators to solve the reduced Einstein's equations. In particular this allows to find solutions by purely algebraic computations.