Quantum curves and conformal field theory

TL;DR

This paper constructs quantum curves associated with algebraic curves using Virasoro algebra representations, connecting matrix models, topological recursion, and conformal field theory concepts.

Contribution

It introduces a unified framework for quantum curves via Virasoro singular vectors, applicable to various algebraic curves and matrix models, extending previous methods.

Findings

Quantum curves correspond to Virasoro singular vectors.

Construction applies to spectral curves of matrix models.

Special cases recover conformal field theory structures.

Abstract

To a given algebraic curve we assign an infinite family of quantum curves (Schr\"odinger equations), which are in one-to-one correspondence with, and have the structure of, Virasoro singular vectors. For a spectral curve of a matrix model we build such quantum curves out of an appropriate representation of the Virasoro algebra, encoded in the structure of the $\alpha/\beta$-deformed matrix integral and its loop equation. We generalize this construction to a large class of algebraic curves by means of a refined topological recursion. We also specialize this construction to various specific matrix models with polynomial and logarithmic potentials, and among other results, show that various ingredients familiar in the study of conformal field theory (Ward identities, correlation functions and a representation of Virasoro operators acting thereon, BPZ equations) arise upon specialization of our formalism to the multi-Penner matrix model.