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

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 Buryak et al. (2018) to this work.