Irregular singularities in Liouville theory

TL;DR

This paper introduces and analyzes irregular singularities in Liouville theory, providing methods to compute conformal blocks and proposing a structure for correlation functions, with implications for Argyres-Douglas theories.

Contribution

It defines irregular singularities in Liouville theory, constructs conformal blocks with these singularities, and predicts partition functions for certain supersymmetric theories.

Findings

Defined bases for conformal blocks with irregular singularities

Developed series expansion techniques for these conformal blocks

Predicted partition functions for Argyres-Douglas theories

Abstract

Motivated by problems arising in the study of N=2 supersymmetric gauge theories we introduce and study irregular singularities in two-dimensional conformal field theory, here Liouville theory. Irregular singularities are associated to representations of the Virasoro algebra in which a subset of the annihilation part of the algebra act diagonally. In this paper we define natural bases for the space of conformal blocks in the presence of irregular singularities, describe how to calculate their series expansions, and how such conformal blocks can be constructed by some delicate limiting procedure from ordinary conformal blocks. This leads us to a proposal for the structure functions appearing in the decomposition of physical correlation functions with irregular singularities into conformal blocks. Taken together, we get a precise prediction for the partition functions of some Argyres-Douglas type theories on the four-sphere.