Four-Dimensional N=2(4) Superstring Backgrounds and The Real Heavens

TL;DR

This paper explores four-dimensional N=2 superstring backgrounds with non-Kähler geometry, linking integrable equations like the Toda and Laplace equations to self-dual Einstein spaces, and demonstrates their T-duality to hyper-Kähler instantons.

Contribution

It introduces new non-Kähler superstring backgrounds characterized by Toda and Laplace equations and connects their integrability to real self-dual Einstein spaces, revealing their geometric and physical significance.

Findings

Constructed a superstring background from Toda equation solutions.

Mapped backgrounds to hyper-Kähler Taub-NUT instantons via T-duality.

Linked integrable equations to properties of real self-dual Einstein spaces.

Abstract

We study N=2(4) superstring backgrounds which are four-dimensional non-\Kahlerian with non-trivial dilaton and torsion fields. In particular we consider the case that the backgrounds possess at least one $U(1)$ isometry and are characterized by the continual Toda equation and the Laplace equation. We obtain a string background associated with a non-trivial solution of the continual Toda equation, which is mapped, under the T-duality transformation, to the hyper-\Kahler Taub-NUT instanton background. It is shown that the integrable property of the non-\Kahlerian spaces have the direct origin in the real heavens: real, self-dual, euclidean, Einstein spaces. The Laplace equation and the continual Toda equation imposed on quasi-\Kahler geometry for consistent string propagation are related to the self-duality conditions of the real heavens with ``translational'' and ``rotational''Killing symmetry respectively.