String Equations for the Unitary Matrix Model and the Periodic Flag Manifold

TL;DR

This paper explores the connection between string equations, the modified Korteweg--de Vries hierarchy, and the symmetric unitary matrix model within the context of the periodic flag manifold, revealing a deep geometric relationship.

Contribution

It establishes a novel link between string equations in matrix models and the geometry of the periodic flag manifold, extending understanding of self--similar solutions in integrable hierarchies.

Findings

Moduli space of solutions is a double cover of the Sato Grassmannian moduli space.

Self--similar solutions correspond to a subset of the Grassmannian moduli space.

The potential mKdV hierarchy relates to a line bundle over the periodic flag manifold.

Abstract

The periodic flag manifold (in the Sato Grassmannian context) description of the modified Korteweg--de Vries hierarchy is used to analyse the translational and scaling self--similar solutions of this hierarchy. These solutions are characterized by the string equations appearing in the double scaling limit of the symmetric unitary matrix model with boundary terms. The moduli space is a double covering of the moduli space in the Sato Grassmannian for the corresponding self--similar solutions of the Korteweg--de Vries hierarchy, i.e. of stable 2D quantum gravity. The potential modified Korteweg--de Vries hierarchy, which can be described in terms of a line bundle over the periodic flag manifold, and its self--similar solutions corresponds to the symmetric unitary matrix model. Now, the moduli space is in one--to--one correspondence with a subset of codimension one of the moduli space in the Sato Grassmannian corresponding to self--similar solutions of the Korteweg--de Vries hierarchy.