Irregular Conformal States and Spectral Curve: Irregular Matrix Model Approach

TL;DR

This paper develops a new formalism for irregular conformal states using irregular vertex operators, linking their inner products to irregular matrix models and analyzing the spectral curve structure across various symmetries.

Contribution

Introduces a novel approach to irregular conformal states via irregular vertex operators and matrix models, avoiding the colliding limit procedure.

Findings

Inner product expressed as irregular matrix model partition function

Spectral curve analyzed for Virasoro, W, and super-symmetries

Loop equations at Nekrasov-Shatashvili limit derived

Abstract

We present recent developments of irregular conformal conformal states. Irregular vertex operators and their adjoint in a new formalism are used to define the irregular conformal states and their inner product instead of using the colliding limit procedure. Free field formalism can be augmented by screening operators which provide more degrees of freedom. The inner product is conveniently given as the partition function of an irregular matrix model. (Deformed) spectral curve is the loop equation of the matrix model at Nekrasov-Shatashivili limit. We present the details of analytic structure of the spectral curve for Virasoso symmetry and its extensions, $W$-symmetry and super-symmetry.