On N=2 supersymmetric gauge theories on S^2XS^2

TL;DR

This paper constructs a supergeometry on S^2XS^2 enabling supersymmetric placement of 4D N=2 gauge theories, analyzes their invariance under a specific superalgebra, and solves BPS equations for localization purposes.

Contribution

It develops a supergeometry framework for N=2 gauge theories on S^2XS^2 and derives exact BPS solutions for localization calculations.

Findings

Supergeometry on S^2XS^2 preserves all supersymmetries.

N=2 gauge theories are invariant under D(2,1,alpha).

Explicit BPS solutions are obtained for localization.

Abstract

We construct a supergeometry based on S^2XS^2 on which four dimensional N=2 gauge theories can be placed supersymmetrically while preserving all supersymmetries. By embedding the supergeometry in four dimensional N=2 supergravity we are able to construct an arbitrary N=2 gauge theory on S^2XS^2. We show that N=2 gauge theories are invariant under the exceptional superalgebra D(2,1,alpha), where alpha is the ratio of the radii of the two S^2's. We solve the supersymmetry fixed points equations for a choice of supercharge in D(2,1,alpha). The solution of these BPS equations, which we find, would serve as the exact saddle point configurations of a localization computation of the partition function of N=2 gauge theories on S^2XS^2.