Algebro-Geometric Solutions of the Generalized Virasoro Constraints

TL;DR

This paper explores algebro-geometric solutions to the KdV hierarchy that satisfy generalized Virasoro constraints, revealing their connection to Virasoro algebra embeddings and testing these solutions against the Witten-Kontsevich tau-function.

Contribution

It introduces a novel class of solutions to the KdV hierarchy satisfying generalized Virasoro constraints and links these solutions to algebraic and geometric structures.

Findings

Solutions are related to Virasoro algebra embeddings.

Connections established with double covers of the projective line.

Links identified with ${ m Gl}(n)$-opers on the punctured disk.

Abstract

We will describe algebro-geometric solutions of the KdV hierarchy whose $\tau$-functions in addition satisfy a generalization of the Virasoro constraints (and, in particular, a generalization of the string equation). We show that these solutions are closely related to embeddings of the positive half of the Virasoro algebra into the Lie algebra of differential operators on the circle. Our results are tested against the case of Witten-Kontsevich $\tau$-function. As by-products, we exhibit certain links of our methods with double covers of the projective line equipped with a line bundle and with ${\rm Gl}(n)$-opers on the punctured disk.