N=2 supersymmetric gauge theories on S2 x S2 and Liouville Gravity

We consider 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 S-2 x S-2 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.