Irregular KZ equations and Kac-Moody representations

TL;DR

This paper constructs irregular representations of affine Kac-Moody algebras and links them to irregular KZ equations, revealing connections to 2D Liouville theory and 4D Argyres-Douglas theories with surface operators.

Contribution

It introduces irregular representations of affine Kac-Moody algebras and derives irregular KZ equations, connecting them to Liouville theory and Argyres-Douglas theories.

Findings

Irregular KZ equations govern conformal blocks with irregular representations.

Connections established between 2D Liouville theory and 4D Argyres-Douglas theories.

Flat connections describe braiding of surface operators on Gaiotto curves.

Abstract

In this paper we construct irregular representations of the affine Kac-Moody algebra $\widehat{sl}(2,\mathbb{C})$. We show how such irregular representations correspond to irregular Gaiotto-Teschner representations of the Virasoro algebra. The intertwiners for such representations satisfy a version of Knizhnik-Zamolodchikov (KZ) equations which we call irregular KZ equations. By connecting to 2d Liouville theory, we show how the conformal blocks governed by our irregular KZ equation correspond to 4d Argyres-Douglas theories with surface operator insertions. The corresponding flat connections describe braiding between such operators on the Gaiotto curve.