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

Abstract

We study 4d type ${\mathcal{H}}_0$ 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 Painlevé 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.

23 pages, typos corrected mainly in appendix (A.1)