Complex Structures, Duality, and WZW-Models in Extended Superspace

TL;DR

The paper analyzes how duality in $(2,2)$ supersymmetric sigma-models with torsion transforms target-space complex structures $J_\pm$ into dual structures $K_\pm$, proving that the space orthogonal to the kernel of $[J_+,J_-]$ is always integrable and thus describable by chiral and twisted chiral coordinates. It shows that even when $[J_+,J_-]$ is nonzero, a substructure with integrable coordinates exists, and explores this via an explicit $SU(2)\times SU(2)$ WZW example, where duality yields noncommuting dual complex structures on $SU(2)\times U(1)$. The authors then construct a new $N=2$ superspace formulation for the $SU(2)\times U(1)$ WZW model using semichiral superfields, providing a concrete realization of noncommuting $K_\pm$ in extended superspace. Together, these results illuminate how extended superspace descriptions and duality interact with complex structures in WZW models and suggest broader applicability of semichiral descriptions in non-Kähler geometries.

Abstract

We find the complex structure on the dual of a complex target space. For $N=(2,2)$ systems, we prove that the space orthogonal to the kernel of the commutator of the left and right complex structures is {\em always} integrable, and hence the kernel is parametrized by chiral and twisted chiral superfield coordinates. We then analyze the particular case of $SU(2)\times SU(2)$, and are led to a new $N=2$ superspace formulation of the $SU(2)\times U(1)$ WZW-model.