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.
This is a review of how sigma models formulated in Superspace have become important tools for understanding geometry. Topics included are: The (hyper)kahler reduction; projective superspace; the generalized Legendre construction; generalized Kahler g
eometry and constructions of hyperkahler metrics on Hermitean symmetric spaces.
I stress how the form of sigma models with (2, 2) supersymmetry differs depending on the number of manifest supersymmetries. The differences correspond to different aspects/formulations of Generalized Kahler Geometry.
We discuss the conditions for extra supersymmetry of the N=(2,2) supersymmetric vector multiplets described in arXiv:0705.3201 [hep-th] and in arXiv:0808.1535 [hep-th]. We find (4,4) supersymmetry for the semichiral vector multiplet but not for the Large Vector Multiplet.
We discuss additional supersymmetries for N = (2, 2) supersymmetric non-linear sigma models described by left and right semichiral superfields.
