Irregular singularities in Liouville theory

TL;DR

This work extends Liouville theory by introducing irregular singularities that arise from collision limits of ordinary vertex operators, defining irregular Virasoro vectors and BPZ-like bases for conformal blocks. It develops multiple complementary constructions—algebraic, free-field, and null-vector PDE approaches—to build and analyze bases for spaces of irregular vectors, and to describe their collision and degeneration limits. The authors propose a Liouville measure for correlators with irregular insertions, enabling a precise prediction for the sphere partition functions of Argyres-Douglas theories via the gauge/CFT correspondence. The results provide a framework connecting extended Teichmüller data, Stokes phenomena, and a gauge-theory dictionary, with significant implications for understanding AD theories and their protected quantities. This work sets the stage for a generalized AGT program that incorporates irregular singularities and their rich boundary structures in the moduli space.

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.

Figures (6)

• Figure 1: The standard graphical representation of a conformal block. In the figure, $\Delta_i$ denotes $\Delta_{\alpha_i}$ and $\delta_i$ denotes $\Delta_{\beta_i}$.
• Figure 2: The graphical representation of a conformal block with two regular punctures, fused in a channel of dimension $\delta$, and a rank $2$ puncture of momentum $\alpha"$, realized inside a rank $1$ channel of momentum $\beta'$ We denote rank $2$ channels with a double arrow. Black dots denote standard or generalized chiral vertex operators.
• Figure 3: The graphical representation of a conformal block where a regular puncture and a rank $1$ puncture are realized inside a rank $1$ channel of momentum $\beta'$.
• Figure 4: The sequence of collision limits which give the conformal blocks of rank $2$.
• Figure 5: Examples of screening paths. From left to right, the unique rank $1$ example, a rank $2$ example with two short paths, another rank $2$ example