Matrix models as conformal field theories: genus expansion

TL;DR

This paper develops a conformal field theory framework for hermitian matrix models, representing their topological expansion through correlation functions of dressed twist fields on hyperelliptic Riemann surfaces, enabling genus-by-genus calculations.

Contribution

It introduces a novel CFT-based approach to analyze matrix models, providing explicit Feynman rules for computing free energy and observables at any genus.

Findings

Derived a topological expansion using CFT on Riemann surfaces

Established a correspondence between branch points and twist fields

Provided a systematic method for genus expansion calculations

Abstract

We obtain the topological expansion of the hermitian matrix model using its representation as a CFT on a hyperelliptic Riemann surface. To each branch point of the Riemann surface we associate an operator which represents a twist field dressed by the modes of the twisted boson. The partition function of the matrix model is computed as a correlation function of such dressed twist fields. The perturbative construction of the dressing operators yields a set of Feynman rules for evaluating the free energy and the loop observables at any genus.