Conformal internal symmetry of $2d$ $\sigma$-models coupled to gravity and a dilaton

TL;DR

This paper uncovers a comprehensive infinite-dimensional symmetry structure in two-dimensional gravity-coupled sigma models, explaining their solution space and spectral parameters through advanced group theoretical methods.

Contribution

It demonstrates the full symmetry algebra as a semi-direct product of affine Kac-Moody and Witt algebras, extending known Lax pairs with a twisted self-duality constraint, and clarifies the role of spectral parameters.

Findings

Full symmetry algebra is a semi-direct product of affine Kac-Moody and Witt algebras.

The extended Lax pair involves a twisted self-duality constraint.

The symmetry acts on off-shell fields and preserves equations of motion.

Abstract

General Relativity reduced to two dimensions possesses a large group of symmetries that exchange classical solutions. The associated Lie algebra is known to contain the affine Kac-Moody algebra $A_1^{(1)}$ and half of a real Witt algebra. In this paper we exhibit the full symmetry under the semi-direct product of $\Lie{A_1^{(1)}}$ by the Witt algebra $\Lie{\Wir}$. Furthermore we exhibit the corresponding hidden gauge symmetries. We show that the theory can be understood in terms of an infinite dimensional potential space involving all degrees of freedom: the dilaton as well as matter and gravitation. In the dilaton sector the linear system that extends the previously known Lax pair has the form of a twisted self-duality constraint that is the analog of the self-duality constraint arising in extended supergravities in higher spacetime dimensions. Our results furnish a group theoretical explanation for the simultaneous occurrence of two spectral parameters, a constant one ($=y$) and a variable one ($=t$). They hold for all $2d$ non-linear $\sigma$-models that are obtained by dimensional reduction of $G/H$ models in three dimensions coupled to pure gravity. In that case the Lie algebra is $\Lie{\Wir \semi G^{(1)}}$; this symmetry acts on a set of off shell fields (in a fixed gauge) and preserves the equations of motion.