Extended supersymmetry of semichiral sigma model in 4D

TL;DR

This paper explores the compatibility of extended supersymmetry with four-dimensional target space torsion in semichiral sigma models, revealing restrictions to Hyperk"ahler geometry and limitations in lifting supersymmetries.

Contribution

It introduces a novel $(1,1)$ superspace formulation for semichiral sigma models in 4D and identifies conditions under which extended supersymmetry can be realized.

Findings

Extended supersymmetry conditions hold for Hyperk"ahler geometries.

Additional supersymmetries in the $SU(2)\otimes U(1)$ WZW model cannot be lifted to $(2,2)$ semichiral formulation.

Restrictions to Hyperk"ahler geometry are confirmed for 4D target spaces with torsion.

Abstract

Briefly: Using a novel $(1,1)$ superspace formulation of semichiral sigma models with $4D$ target space, we investigate if an extended supersymmetry in terms of semichirals is compatible with having a $4D$ target space with torsion. In more detail: Semichiral sigma models have $(2,2)$ supersymmetry and Generalized K\"ahler target space geometry by construction. They can also support $(4,4)$ supersymmetry and Generalized Hyperk\"ahler geometry, but when the target space is four dimensional indications are that the geometry is restricted to Hyperk\"ahler. To investigate this further, we reduce the model to $(1,1)$ superspace and construct the extra (on-shell) supersymmetries there. We then find the conditions for a lift to $(2,2)$ super space and semichiral fields to exist. Those conditions are shown to hold for Hyperk\"ahler geometries. The $SU(2)\otimes U(1)$ WZW model, which has $(4,4)$ supersymmetry and a semichiral description, is also investigated. The additional supersymmetries are found in $(1,1)$ superspace but shown {\em not} to be liftable to a $(2,2)$ semichiral formulation.