$\mathcal{N}=2$ supersymmetric gauge theories on $S^2\times S^2$ 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 $S^2\times S^2$ 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.

48+1 pages, no figures; v2. typos corrected, comments and refs added; v3 JHEP version

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• N=2 supersymmetric gauge theory on connected sums of $S^2\times S^2$
• Partition functions for equivariantly twisted $\mathcal{N}=2$ gauge theories on toric Kähler manifolds