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 × 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.
We study holomorphic blocks in the three dimensional N = 2 gauge theory that describes the CP 1 model. We apply exact WKB methods to analyze the line operator identities associated to the holomorphic blocks and derive the analytic continuation formulae of the blocks as the twisted mass and FI parameter are varied. The main technical result we utilize is the connection formula for the 1 φ 1 q-hypergeometric function. We show in detail how the q-Borel resummation methods reproduce the results obtained previously by using block-integral methods.
We compute the exact path integral of N = 2 supersymmetric gauge theories with general gauge group on RP 4 and a Z 2 -quotient of the hemi-S 4 . By specializing to SU (2) superconformal quivers, we show that these, together with hemi-S 4 partition functions, compute Liouville correlators on unoriented/open Riemann surfaces. We perform explicit checks for Riemann surfaces obtained as Z 2 quotients of the sphere and the torus. We also discuss the coupled 3d − 4d systems associated to Liouville amplitudes with boundary punctures.
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