AGT/Z$_2$

TL;DR

This paper establishes a connection between Liouville/Toda conformal field theories on Riemann surfaces with boundaries and cross-caps and supersymmetric observables in four-dimensional N=2 gauge theories, extending the AGT correspondence to nonorientable spaces.

Contribution

It introduces a novel framework relating 2D CFT correlators with 4D N=2 gauge theories on Z2 quotients, including boundary and cross-cap states, and tests this via partition functions on nonorientable manifolds.

Findings

Derived RP^4 partition functions from 3d and 4d components.

Reproduced RP^2 partition functions to verify dualities.

Computed boundary and cross-cap wavefunctions in Toda CFT.

Abstract

We relate Liouville/Toda CFT correlators on Riemann surfaces with boundaries and cross-cap states to supersymmetric observables in four-dimensional N=2 gauge theories. Our construction naturally involves four-dimensional theories with fields defined on different Z$_2$ quotients of the sphere (hemisphere and projective space) but nevertheless interacting with each other. The six-dimensional origin is a Z$_2$ quotient of the setup giving rise to the usual AGT correspondence. To test the correspondence, we work out the RP$^4$ partition function of four-dimensional N=2 theories by combining a 3d lens space and a 4d hemisphere partition functions. The same technique reproduces known RP$^2$ partition functions in a form that lets us easily check two-dimensional Seiberg-like dualities on this nonorientable space. As a bonus we work out boundary and cross-cap wavefunctions in Toda CFT.