$\mathcal{N}=2$ supersymmetric gauge theories on $S^2\times S^2$ and   Liouville Gravity
  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. 
Aditya Bawane, Giulio Bonelli, Massimiliano Ronzani and Alessandro   Tanzini
hep-th math-ph math.AG math.MP math.RT