Unitary One Matrix Models: String Equations and Flows

TL;DR

This paper reviews symmetric unitary one matrix models, focusing on string equations, mKdV flows, and Virasoro constraints, and explores their connections to the Sato Grassmannian and fermionic Fock space formalism.

Contribution

It provides a detailed analysis of the solution space of the string equation within the framework of the Sato Grassmannian and fermionic formalism, highlighting new connections.

Findings

Explicit description of the solution space to the string equation

Connection between flows and the Sato Grassmannian via Plucker embedding

Invariant subspace under flows in the solution space

Abstract

We review the Symmetric Unitary One Matrix Models. In particular we discuss the string equation in the operator formalism, the mKdV flows and the Virasoro Constraints. We focus on the $\t$-function formalism for the flows and we describe its connection to the (big cell of the) Sato Grassmannian $\Gr$ via the Plucker embedding of $\Gr$ into a fermionic Fock space. Then the space of solutions to the string equation is an explicitly computable subspace of $\Gr\times\Gr$ which is invariant under the flows.