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.
Highlights
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
When subjected to the same test they fail to satisfy some of the conditions. This leads to the surprising conclusion that (4, 4) supersymmetry in a (1, 1) formulation of a (2, 2) sigma model with on-shell supersymmetry is incompatible with the introduction of the (2, 2) auxiliary fields
We have extended the (1, 1) formulation of semichiral sigma models to allow for a treatment of extra super symmetries with on-shell closure
Summary
It is an interesting fact that on-shell closure of the algebra, together with conditions that come from invariance of the action, requires that the function c(XL, XR) defined by (2.11) is constant with absolute value less than one, which means that the geometry is hyperkahler. This on-shell closure is different than the one which arises in the general (1, 1) discussion of extended susy [8] which locates the nonclosure of the algebra to the (+, −) sector where the commutator [J(+), J(−)] multiplies the field equation. In our (1, 1) language, the relation to semichirals is given by (3.5) in (XL, XR) coordinates. We denote a generic complex structure by
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