Additional symmetries of KP, Grassmannian, and the string equation II

TL;DR

This paper explores additional symmetries in matrix integrable hierarchies, revealing their actions on Grassmannians and deriving Virasoro constraints, thus deepening understanding of matrix models and string equations.

Contribution

It extends previous work by analyzing symmetries in multi-component KP, KdV, and modified KdV hierarchies, and their effects on Grassmannian and tau-functions.

Findings

Action of additional symmetries on Grassmannian derived

Virasoro constraints on tau-functions established

Matrix string equations linked to matrix models

Abstract

As in the first part of this paper (hep-th 9204092), solutions to a string equation are regarded as fixed points of some additional symmetries of a hierarchy of integrable equations. In this part matrix hierarchies are studied: the multi-component KP and KdV hierarchies, and the modified KdV hierarchy as their reduction. In particular, the action of additional symmetries on the Grassmannian is found, as well as Virasoro constraints on the $\tau$-functions. The matrix string equations are known to be involved in some matrix models.