Space Time Interpretation of S-Duality and Supersymmetry Violations of T-Duality

TL;DR

This paper explores how S-duality and T-duality transformations in string theory affect four-dimensional backgrounds, revealing their impact on gravitational solutions, instantons, and supersymmetry violations.

Contribution

It demonstrates the space-time interpretation of S-duality, maps gravitational instantons via dualities, and identifies conditions under which supersymmetry is broken during T-duality transformations.

Findings

S-duality induces an $SL(2,R)$ transformation matching the Ehlers-Geroch group.

ALE instantons map to Taub-NUT backgrounds under dualities.

Self-duality of metrics is generally not preserved unless specific conditions are met.

Abstract

The S--duality transformations of the lowest order string effective theory admit a space time interpretation for 4-dim backgrounds with one Killing symmetry. Starting from pure gravity and performing a sequence of intertwined T-S-T duality transformations we obtain new solutions which are always pure gravitational. In this fashion, S-duality induces an $SL(2,R)$ transformation in the space of target space metrics which coincides with the action of the Ehlers-Geroch group and interchanges the electric with the magnetic aspects of gravity. Specializing to gravitational instanton backgrounds we show that ALE instantons are mapped to (multi) Taub-NUT backgrounds and vice-versa. We find, however, that the self-duality of the metric is not generically preserved, unless the corresponding Killing vector field has self-dual covariant derivatives. Thus, the T-S-T transformations are not always compatible with the world-sheet supersymmetry of $N=4$ superconformal string vacua. We also provide an algebraic characterization of the corresponding obstruction and associate it with a breakdown of space time supersymmetry under rotational T-duality transformations.