N = 2 $$ \mathcal{N}=2 $$ supersymmetric gauge theories on S2 × S2 and Liouville Gravity

Abstract

We consider $$ \mathcal{N}=2 $$ supersymmetric gauge theories on four manifolds admitting an isometry. Generalized Killing spinor equations are derived from the consistency of supersymmetry algebrae and solved in the case of four manifolds admitting a U(1) isometry. This is used to explicitly compute the supersymmetric path integral on S2 × S2 via equivariant localization. The building blocks of the resulting partition function are shown to contain the three point functions and the conformal blocks of Liouville Gravity.