Accessory Parameter of Confluent Heun Equations, Voros Periods and classical irregular conformal blocks

TL;DR

This paper derives series expansions for accessory parameters of confluent Heun equations using Voros periods, compares them with classical conformal blocks, and explores a conjecture linking them, including irregular singularities.

Contribution

It provides a detailed method to connect Voros periods with classical conformal blocks for confluent Heun equations, extending previous work to irregular singularities.

Findings

Derived formal series expansions for accessory parameters.

Compared expansions with classical conformal blocks.

Examined the conjectured relationship including irregular cases.

Abstract

For the Heun differential equation and all of its confluent equations, we derive formal series expansions of the accessory parameters using the Voros periods. We then compare these expansions with the classical conformal blocks recently obtained by Bonelli--Shchechkin--Tanzini, and examine the Zamolodchikov-type conjecture expected to hold between them, allowing for irregular singularities. In particular, as an extension of the previous works of Mironov--Morozov, Piatek--Pietrykowski and Lisovyy--Naidiuk, we provide a detailed prescription for choosing cycles on the spectral curve that yield the Voros period which corresponds to the classical (regular or irregular) conformal blocks through the accessory parameter.