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

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.