Additional symmetries of KP, Grassmannian, and the string equation

TL;DR

This paper reveals that solutions to the string equation are generated by Grassmannian elements invariant under the infinitesimal operators of the additional symmetries of the KdV hierarchy, extending to arbitrary orders.

Contribution

It demonstrates that the differential operator generating solutions is the infinitesimal operator of the KdV additional symmetries group, generalizing previous results to all KdV orders.

Findings

Solutions are generated by Grassmannian elements invariant under additional symmetries.

The operator is identified as the infinitesimal generator of the symmetry group.

Virasoro constraints are derived in a more general form.

Abstract

It is well-known that solutions to the string equation are generated by elements of Sato's Grassmannian which are invariant under action of some differential operator. Here it is shown that this operator is nothing else than the infinitesimal operator of the group of additional symmetries of the KdV flow. This is done for KdV hierarchies of arbitrary orders. Virasoro constraints are obtained in a slightly more general form than they are usually written.