# Argyres-Douglas theories and Liouville Irregular States

## TL;DR

This paper explores irregular states in Liouville theory to compute partition functions of Argyres-Douglas theories, revealing connections to Painlevé equations and symmetry invariances.

## Contribution

It introduces a method to evaluate irregular conformal blocks for Argyres-Douglas theories and links them to Painlevé equations and Weyl group symmetries.

## Key findings

- Reproduces strong coupling expansions via irregular conformal blocks.
- Suggests a relation between rank 3 irregular states and Painlevé II.
- Demonstrates invariance under Weyl group actions.

## Abstract

We study irregular states of rank-two and three in Liouville theory, based on an ansatz proposed by D. Gaiotto and J. Teschner. Using these irregular states, we evaluate asymptotic expansions of irregular conformal blocks corresponding to the partition functions of $(A_1,A_3)$ and $(A_1,D_4)$ Argyres-Douglas theories for general $\Omega$-background parameters. In the limit of vanishing Liouville charge, our result reproduces strong coupling expansions of the partition functions recently obtained via the Painlev\'e/gauge correspondence. This suggests that the irregular conformal block for one irregular singularity of rank 3 on sphere is also related to Painlev\'e II. We also find that our partition functions are invariant under the action of the Weyl group of flavor symmetries once four and two-dimensional parameters are correctly identified. We finally propose a generalization of this parameter identification to general irregular states of integer rank.