Semichirals and Four Dimensional Geometry

TL;DR

This paper investigates semichiral sigma models with four-dimensional targets, revealing that extended supersymmetry manifests only in hyperkähler geometries, with specific models serving as notable exceptions.

Contribution

It introduces a new $(1,1)$ superspace formulation for semichiral models, clarifies conditions for extended supersymmetry, and analyzes the limitations of semichiral realizations in $(2,2)$ superspace.

Findings

Manifest extended supersymmetry requires hyperkähler geometry.

The $SU(2)\otimes U(1)$ WZW model's extra supersymmetries cannot be realized in semichiral $(2,2)$ superspace.

Semichiral models with four-dimensional targets are constrained by geometric conditions for supersymmetry.

Abstract

Semichiral $(2,2)$ sigma models with $4d$ target space are discussed. A novel description in $(1,1)$ superspace allows an analysis of possible extended supersymmetries. It is argued that a manifest semichiral realization of an extra supersymmetry is only possible for hyperk\"ahler target geometry. A semichiral formulation of the $SU(2)\otimes U(1)$ WZW model is seemingly a counterexample to this. After deriving the extra supersymmetries of this model in $(1,1)$ superspace it is shown that they cannot be lifted to transformations of semichirals in $(2,2)$ superspace.