Irregular conformal block, spectral curve and flow equations

TL;DR

This paper explores the classical limit of irregular conformal blocks related to Argyres-Douglas theories, using spectral curves and flow equations derived from matrix models and symmetry considerations.

Contribution

It introduces a method to analyze irregular conformal blocks via spectral curves and flow equations, connecting them to gauge theories and the AGT conjecture.

Findings

Spectral curve reduces to second and third order differential equations.

Flow equations determine the irregular conformal block.

Connection established between spectral curves and Argyres-Douglas partition functions.

Abstract

Irregular conformal block is motivated by the Argyres-Douglas type of N=2 super conformal gauge theory. We investigate the classical/NS limit of the irregular conformal block using spectral curve on a Riemann surface with irregular punctures, which is equivalent to the loop equation of irregular matrix model. The spectral curve is reduced to the second order (Virasoro symmetry, $SU(2)$ for the gauge theory) and third order ($W_3$ symmetry, $SU(3)$) differential equations of a polynomial with finite degree. The Virasoro and W symmetry generate flow equations in the spectral curve and determine the irregular conformal block, hence the partition function of the Argyres-Douglas theory ala AGT conjecture.