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.

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 show that the minimal gravity of Lee-Yang series on disk is a solution to the open KdV hierarchy proposed for the intersection theory on the moduli space of Riemann surfaces with boundary.

We consider timelike Liouville theory with FZZT-like boundary conditions. The bulk one-point and boundary two-point structure constants on a disk are derived using bootstrap. We find that these structure constants are not the analytic continuations of their spacelike counterparts.

Liouville field theory approach to 2-dimensional gravity possesses the duality (b ↔ b −1 ). The matrix counterpart of minimal gravity M(q, p) (q < p co-prime) is effectively described on A q−1 Frobenius manifold, which may exhibit a similar duality p ↔ q, and allow a description on A p−1 Frobenius manifold. We have positive results from the bulk one-point and the bulk-boundary two-point correlations on disk that the dual description of the Frobenius manifold works for the unitary series M(q, q + 1). However, for the Lee-Yang series M(2, 2q + 1) on disk the duality is checked only partially. The main difficulty lies in the absence of a canonical description of trace in the continuum limit.

In this note we provide proofs of various expressions for expectation values of symmetric polynomials in \betaβ-deformed eigenvalue models with quadratic, linear, and logarithmic potentials. The relations we derive are also referred to as superintegrability. Our work completes proofs of superintegrability statements conjectured earlier in literature.