Conformal Field Theory Techniques in Random Matrix models

TL;DR

This paper demonstrates how conformal field theory techniques can be applied to random matrix models, specifically hermitian matrices, to facilitate explicit calculations of spectral properties using free fermion and bosonic formalisms.

Contribution

It introduces an operator construction of the collective field theory on a hyperelliptic Riemann surface, enabling explicit computation of spectral kernels and eigenvalue probabilities.

Findings

Spectral kernel expressions valid at macroscopic and microscopic scales.

Explicit operator formalism for hermitian matrix models.

Framework adaptable to various generalizations.

Abstract

In these notes we explain how the CFT description of random matrix models can be used to perform actual calculations. Our basic example is the hermitian matrix model, reformulated as a conformal invariant theory of free fermions. We give an explicit operator construction of the corresponding collective field theory in terms of a bosonic field on a hyperelliptic Riemann surface, with special operators associated with the branch points. The quasiclassical expressions for the spectral kernel and the joint eigenvalue probabilities are then easily obtained as correlation functions of current, fermionic and twist operators. The result for the spectral kernel is valid both in macroscopic and microscopic scales. At the end we briefly consider generalizations in different directions.