$\mathcal{N}=2$ gauge theories on unoriented/open four-manifolds and their AGT counterparts

TL;DR

This paper computes exact partition functions of $ =2$ supersymmetric gauge theories on unoriented four-manifolds and relates them to Liouville correlators on Riemann surfaces with boundaries, extending the AGT correspondence.

Contribution

It extends the AGT correspondence to unoriented and open four-manifolds, providing explicit computations and checks for Liouville correlators with boundary conditions.

Findings

Exact partition functions on $ ext{RP}^4$ and hemi-$S^4$ computed.

Liouville correlators on unoriented Riemann surfaces related to gauge theory.

Explicit checks for $ ext{Z}_2$ quotients of sphere and torus.

Abstract

We compute the exact path integral of $\mathcal{N}=2$ supersymmetric gauge theories with general gauge group on $\mathbb{RP}^4$ and a $\mathbb{Z}_2$-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 $\mathbb{Z}_2$ quotients of the sphere and the torus. We also discuss the coupled $3d\!-\!4d$ systems associated to Liouville amplitudes with boundary punctures.