Virasoro constraints revisited

TL;DR

This paper revisits the Virasoro constraints, explores their connection to Hirota bilinear equations, and provides solutions for non-homogeneous cases arising from matrix models with boundary conditions, including positive eigenvalue Hermitean matrices.

Contribution

It offers new insights into Virasoro constraints, linking them to Hirota equations and solving non-homogeneous cases with boundary conditions in matrix models.

Findings

Established the relation between Virasoro constraints and Hirota bilinear equations.

Provided a method to solve non-homogeneous Virasoro constraints for boundary matrix models.

Demonstrated solutions for Hermitean matrices with positive eigenvalues.

Abstract

We revisit the Virasoro constraints and explore the relation to the Hirota bilinear equations. We furthermore investigate and provide the solution to non-homogeneous Virasoro constraints, namely those coming from matrix models whose domain of integration has boundaries. In particular, we provide the example of Hermitean matrices with positive eigenvalues in which case one can find a solution by induction on the rank of the matrix model.