Closed extended $r$-spin theory and the Gelfand-Dickey wave function

TL;DR

This paper introduces a generalized genus-zero r-spin theory with a negative-one twist insertion, linking its generating function to the Gelfand-Dickey hierarchy's wave function, thus extending the understanding of r-spin theories.

Contribution

It establishes a connection between the closed extended r-spin theory's generating function and the Gelfand-Dickey hierarchy's wave function at genus zero, a novel generalization.

Findings

The genus-zero closed extended r-spin generating function matches the Gelfand-Dickey wave function.

The theory generalizes previous r-spin models by including a negative-one twist insertion.

Results parallel those found in open r-spin theory, broadening the theoretical framework.

Abstract

We study a generalization of genus-zero $r$-spin theory in which exactly one insertion has a negative-one twist, which we refer to as the "closed extended" theory, and which is closely related to the open $r$-spin theory of Riemann surfaces with boundary. We prove that the generating function of genus-zero closed extended intersection numbers coincides with the genus-zero part of a special solution to the system of differential equations for the wave function of the $r$-th Gelfand-Dickey hierarchy. This parallels an analogous result for the open $r$-spin generating function in the companion paper to this work.