# Resurgence, Painleve Equations and Conformal Blocks

## TL;DR

This paper explores the resurgent structure of Painleve equations and their conformal block expansions, revealing intricate relations across different sectors and critical points, with applications in statistical physics and matrix models.

## Contribution

It demonstrates the physical implications of resurgence in Painleve equations through conformal blocks, connecting various non-perturbative sectors and phase transition physics.

## Key findings

- Resurgence links different instanton sectors and critical points.
- Applications to 2D Ising model, matrix models, and fermion statistics.
- Resurgent structures exist for both convergent and divergent expansions.

## Abstract

We discuss some physical consequences of the resurgent structure of Painleve equations and their related conformal block expansions. The resurgent structure of Painleve equations is particularly transparent when expressed in terms of physical conformal block expansions of the associated tau functions. Resurgence produces an intricate network of inter-relations; some between expansions around different critical points, others between expansions around different instanton sectors of the expansions about the same critical point, and others between different non-perturbative sectors of associated spectral problems, via the Bethe-gauge and Painleve-gauge correspondences. Resurgence relations exist both for convergent and divergent expansions, and can be interpreted in terms of the physics of phase transitions. These general features are illustrated with three physical examples: correlators of the 2d Ising model, the partition function of the Gross-Witten-Wadia matrix model, and the full counting statistics of one dimensional fermions, associated with Painleve VI, Painleve III and Painleve V, respectively.