New Algebraic Structures from Hermitian One-Matrix Model

TL;DR

This paper introduces a new algebraic structure derived from the Hermitian one-matrix model, extending the Virasoro algebra with additional parameters and central extension, using conformal field theory methods.

Contribution

It constructs a generalized Virasoro constraint from multi-loop equations and enlarges it with a central extension, revealing a new algebraic structure.

Findings

Derived a generalized Virasoro algebra from multi-loop equations.

Enlarged the algebra with a central extension, creating a new algebraic framework.

Provided a bosonic realization of the subalgebra.

Abstract

Virasoro constraint is the operator algebra version of one-loop equation for a Hermitian one-matrix model, and it plays an important role in solving the model. We construct the realization of the Virasoro constraint from the Conformal Field Theory (CFT) method. From multi-loop equations of the one-matrix model, we get a more general constraint. It can be expressed in terms of the operator algebras, which is the Virasoro subalgebra with extra parameters. In this sense, we named as generalized Virasoro constraint. We enlarge this algebra with central extension, this is a new kind of algebra, and the usual Virasoro algebra is its subalgebra. And we give a bosonic realization of its subalgebra.