A note on rank $\frac{3}{2}$ Liouville irregular block

Abstract

This paper focuses on a conformal block with rank $\frac{3}{2}$ irregular singularity which corresponds to the prepotential of the ${\mathcal{H}}_1$ Argyres-Douglas theory in $\Omega$ background. We derive this irregular conformal block using generalized holomorphic anomaly recursion relation. This results is an expression which is a power series in $\Omega$-background parameters ${\epsilon }_{1,2}$ and exact in coupling. We have verified that in small coupling regime our result is consistent with previously known expressions. Furthermore we derive the Deformed Seiberg-Witten curve which provides an alternative tool to explore above mentioned theory in Nekrasov-Shatashvili limit of $\Omega$-background. We checked that the results are in complete agreement with the holomorphic anomaly approach.

19 pages, a reference added, typos corrected