A note on rank 5/2 Liouville irregular block, Painlev\'e 1 and the ${\cal H}_0$ Argyres-Douglas theory

TL;DR

This paper explores the connection between Liouville irregular states, Painlevé I tau-functions, and the ${ m H}_0$ Argyres-Douglas theory within the Omega-background, providing new insights into their interrelations and computational approaches.

Contribution

It constructs a rank 5/2 Liouville irregular state for the ${ m H}_0$ Argyres-Douglas theory and compares results with the Holomorphic anomaly and Painlevé I tau-function, offering novel computational methods.

Findings

Agreement with generalized Holomorphic anomaly expansion

Confirmation of Painlevé I tau-function correspondence in self-dual case

Analysis of Nekrasov-Shatashvili limit via WKB and deformed Seiberg-Witten curve

Abstract

We study 4d type ${\cal H}_0$ Argyres-Douglas theory in $\Omega$-background by constructing Liouville irregular state of rank 5/2. The results are compared with generalized Holomorphic anomaly approach, which provides order by order expansion in $\Omega$-background parameters $\epsilon_{1,2}$. Another crucial test of our results provides comparison with respect to Painlev\'{e} 1 $\tau$-function, which was expected to be hold in self-dual case $\epsilon_1=-\epsilon_2$. We also discuss Nekrasov-Shatashvili limit $\epsilon_1=0$, accessible either by means of deformed Seiberg-Witten curve, or WKB methods.