On the Relation of Four-Dimensional N=2,4 -- Supersymmetric String Backgrounds to Integrable Models

TL;DR

This paper explores how certain four-dimensional N=2 supersymmetric string backgrounds relate to integrable models, revealing solutions linked to the Liouville and Toda equations, and connecting to known gravitational instantons like Eguchi-Hanson.

Contribution

It demonstrates the connection between supersymmetric string backgrounds and integrable equations, providing explicit solutions and dualities involving gravitational instantons.

Findings

Supersymmetric backgrounds correspond to solutions of Liouville and Toda equations.

Duality transformations relate these solutions to Eguchi-Hanson instantons.

Non-Kählerian backgrounds with specific symmetries arise from integrable models.

Abstract

In this letter we discuss the relation of four-dimensional, $N=2$ supersymmetric string backgrounds to integrable models. In particular we show that non-K\"ahlerian gravitational backgrounds with one $U(1)$ isometry plus non-trivial antisymmetric tensor and dilaton fields arise as the solutions of the Liouville equation or, for the case of vanishing central charge deficit, as the solutions of the continual Toda equation. When performing an Abelian duality transformation, a particular class of solutions of the continual Toda equation leads to the well-known gravitational Eguchi-Hanson instanton background with self-dual curvature tensor.