Accessory parameters in confluent Heun equations and classical irregular conformal blocks

TL;DR

This paper explores the connection between accessory parameters in confluent Heun equations and classical irregular conformal blocks, extending known relations to new classes involving irregular vertex operators and gauge theory correspondences.

Contribution

It introduces a novel class of accessory parameter functions linked to all-order Bohr-Sommerfeld periods and relates them to irregular conformal blocks and gauge theories.

Findings

Extended the relation between accessory parameters and conformal blocks to confluent cases.

Connected irregular conformal blocks with gauge theory partition functions.

Proposed conjectural correspondence with Argyres-Douglas theories.

Abstract

Classical Virasoro conformal blocks are believed to be directly related to accessory parameters of Floquet type in the Heun equation and some of its confluent versions. We extend this relation to another class of accessory parameter functions that are defined by inverting all-order Bohr-Sommerfeld periods for confluent and biconfluent Heun equation. The relevant conformal blocks involve Nagoya irregular vertex operators of rank 1 and 2 and conjecturally correspond to partition functions of a 4D $\mathcal{N}=2$, $N_f=3$ gauge theory at strong coupling and an Argyres-Douglas theory.