Extended supersymmetry of semichiral sigma model in 4D

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 Kahler target space geometry by construction. They can also support $(4,4)$ supersymmetry and Generalized Hyperkahler geometry, but when the target space is four dimensional indications are that the geometry is restricted to Hyperkahler. 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 Hyperkahler 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.