Classical Yang-Baxter Equation from Supergravity

TL;DR

This paper extends the open-closed string map to generate solutions in generalized supergravity, linking the Classical Yang-Baxter equation with gravity and broadening the scope of Yang-Baxter deformations beyond coset spaces.

Contribution

It introduces a gravity-based method to classify r-matrix solutions to the CYBE, generalizing Yang-Baxter deformations to non-coset spaces.

Findings

Reproduces the Classical Yang-Baxter equation from supergravity equations of motion.

Identifies the most general r-matrix solutions for the CYBE.

Provides a gravity-based classification of r-matrix solutions.

Abstract

We promote the open-closed string map, originally formulated by Seiberg \& Witten, to a solution generating prescription in generalized supergravity. The approach hinges on a knowledge of an antisymmetric bivector $\Theta$, built from antisymmetric products of Killing vectors, which is specified by the equations of motion. In the cases we study, the equations of motion reproduce the Classical Yang-Baxter equation (CYBE) and $\Theta$ is the most general $r$-matrix solution. Our work generalizes Yang-Baxter deformations to non-coset spaces and unlocks gravity as a means to classify $r$-matrix solutions to the CYBE.