The Geometry of $N=2$ Strings with Torsion

TL;DR

This paper explores the geometric structures of $N=2$ string theories with torsion, revealing how different superfield choices lead to diverse space-time geometries and dynamics, including anti-self-dual curvature and potential-driven metrics.

Contribution

It introduces a new class of $N=2$ string models with torsion and derives their associated geometric and dynamical properties, including an action for the potential $K$ and conditions for conformal invariance.

Findings

Space-time can be Kahler or hermitian with torsion depending on superfield content.

The string spectrum includes a scalar potential $K$ that determines geometry.

Conditions for conformal invariance involve vanishing of the lowest order conformal anomaly.

Abstract

$N=2$ string theories are formulated in space-times with 2 space and 2 time dimensions. If the world-sheet matter system consists of 2 chiral superfields, the space-time is Kahler and the dynamics are those of anti-self-dual gravity. If instead one chiral superfield and one twisted chiral superfield are used, the space-time is a hermitian manifold with torsion and a dilaton. The string spectrum consists of a scalar, which is a potential $K$ determining the metric, torsion and dilaton. The dynamics imply that the curvature with torsion is anti-self-dual, and an action is found for the potential $K$. It is argued that any $N=2$ sigma-model with twisted chiral multiplets in any dimension can be deformed to a conformally invariant theory if the lowest order contribution to the conformal anomaly vanishes. If there are isometries, more general geometries are possible in which the dilaton is the Killing potential for a holomorphic Killing vector.