Actions For (2,1) Sigma-Models and Strings

TL;DR

This paper derives effective actions for (2,0) and (2,1) superstrings through sigma-model analysis, revealing generalized geometries with torsion, self-dual curvature, and extended supersymmetry structures across various signatures.

Contribution

It introduces a generalized geometric framework for superstring sigma-models, incorporating torsion, self-duality, and extended supersymmetry, expanding the understanding of superstring effective actions.

Findings

Curvature with torsion is self-dual in four dimensions.

Sigma-models exhibit (4,1) supersymmetry with hyperkähler structure in Euclidean signature.

The theory features a twisted superconformal algebra with SL(2,R) Kac-Moody symmetry.

Abstract

Effective actions are derived for (2,0) and (2,1) superstrings by studying the corresponding sigma-models. The geometry is a generalisation of Kahler geometry involving torsion and the field equations imply that the curvature with torsion is self-dual in four dimensions, or has SU(n,m) holonomy in other dimensions. The Yang-Mills fields are self-dual in four dimensions and satisfy a form of the Uhlenbeck-Yau equation in higher dimensions. In four dimensions with Euclidean signature, there is a hyperkahler structure and the sigma-model has (4,1) supersymmetry, while for signature (2,2) there is a hypersymplectic structure consisting of a complex structure squaring to -1 and two real structures squaring to 1. The theory is invariant under a twisted form of the (4,1) superconformal algebra which includes an SL(2,R) Kac-Moody algebra instead of an SU(2) Kac-Moody algebra. Kahler and related geometries are generalised to ones involving real structures.