$\mathcal{N}=2$ supersymmetric gauge theories on $S^2\times S^2$ and Liouville Gravity

TL;DR

This work develops a framework for placing ${ m N}=2$ supersymmetric gauge theories on curved 4-manifolds with a $U(1)$ isometry, deriving generalized Killing spinor equations and solving them on ${S^2 imes S^2}$. By performing an equivariant twist and reexpressing the theory in twisted variables, the authors localize the path integral to fixed points of the isometry, gluing Nekrasov partition functions across toric patches and integrating over Cartan moduli. Remarkably, Liouville Gravity data—specifically three-point functions and conformal blocks—emerge as building blocks of the partition function, linking 4D gauge theory to LG via an AGT-like structure and M5-brane anomaly considerations. The results illuminate holomorphic factorization properties, organize the computation in a toric-geometric language, and suggest rich avenues for further connections to Gromov-Witten/Donaldson invariants, surface operators, and holographic duals.

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.