Integrable deformations of dimensionally reduced gravity

TL;DR

This paper develops two new families of integrable deformations of dimensionally reduced gravity in two dimensions, preserving the Lax structure and integrability, including Hamiltonian integrability for one deformation.

Contribution

It introduces two novel deformation methods for 2D reduced gravity that maintain integrability and the Lax structure, extending previous models and embedding Yang-Baxter deformations.

Findings

Constructed flat Lax representations for both deformations.

Proved Hamiltonian integrability for the Auxiliary Field Deformation.

Extended the class of integrable models in 2D gravity.

Abstract

Dimensional reduction of gravity theories to $D=2$ along commuting Killing isometries is well-known to be classically integrable. The resulting system typically features a coset $\sigma$-model coupled to a dilaton and a scale factor of the dimensional reduction. In this article, we construct two families of deformations of dimensionally reduced gravity that preserve the Lax integrable structure. The first family is an extension of the Auxiliary Field Deformation recently introduced by Ferko and Smith, while the second family consists in the embedding of the Yang-Baxter $\sigma$-model into $D=2$ dimensionally reduced gravity. For both deformations we construct flat Lax representations. The Auxiliary Field Deformation, in particular, preserves the rich algebraic structure underlying the undeformed model and, leaving the canonical structure of the Lax connection's spatial components essentially unchanged, allows us to prove its integrability also in the Hamiltonian sense.