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 $S^2 \times S^2$ 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.

%B Journal of High Energy Physics %V 2015 %P 54 %8 Jul %G eng %U https://doi.org/10.1007/JHEP07(2015)054 %R 10.1007/JHEP07(2015)054