Classical Virasoro irregular conformal block

TL;DR

This paper derives the classical Virasoro irregular conformal block of arbitrary rank using two methods: the beta-deformed irregular matrix model and the generalized Mathieu equation, connecting conformal blocks with integrable systems.

Contribution

It provides a novel derivation of the classical Virasoro irregular conformal block for arbitrary rank via matrix models and differential equations, unifying different approaches.

Findings

Explicit form of the irregular conformal block obtained

Connection established between matrix model and Mathieu equation

Results applicable in the classical or Nekrasov-Shatashvili limit

Abstract

Virasoro irregular conformal block with arbitrary rank is obtained for the classical limit or equivalently Nekrasov-Shatashvili limit using the beta-deformed irregular matrix model (Penner-type matrix model for the irregular conformal block). The same result is derived using the generalized Mathieu equation which is equivalent to the loop equation of the irregular matrix model.