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

TL;DR

This paper derives a new irregular conformal block related to Argyres-Douglas theory using holomorphic anomaly recursion, providing exact power series expressions and a deformed Seiberg-Witten curve, verified against known results.

Contribution

It introduces a novel derivation of the rank 3/2 irregular conformal block via holomorphic anomaly recursion and constructs the deformed Seiberg-Witten curve for the theory.

Findings

Derived the irregular conformal block as a power series in Omega-background parameters.

Verified the results are consistent with known expressions in the small coupling regime.

Constructed the deformed Seiberg-Witten curve matching holomorphic anomaly approach.

Abstract

This paper focuses on a conformal block with rank $\frac{3}{2}$ irregular singularity which corresponds to the prepotential of the ${\cal H}_1$ 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.