We conclude the construction of $r$-spin theory in genus zero for Riemann surfaces with boundary. In particular, we define open $r$-spin intersection numbers, and we prove that their generating function is closely related to the wave function of the
$r$th Gelfand--Dickey integrable hierarchy. This provides an analogue of Wittens $r$-spin conjecture in the open setting and a first step toward the construction of an open version of Fan--Jarvis--Ruan--Witten theory. As an unexpected consequence, we establish a mysterious relationship between open $r$-spin theory and an extension of Wittens closed theory.
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 bounda
ry. 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.
We show that the vanishing of the $(g+1)$-st power of the theta divisor on the universal abelian variety $mathcal{X}_g$ implies, by pulling back along a collection of Abel--Jacobi maps, the vanishing results in the tautological ring of $mathcal{M}_{g
,n}$ of Looijenga, Ionel, Graber--Vakil, and Faber--Pandharipande. We also show that Pixtons double ramification cycle relations, which generalize the theta vanishing relations and were recently proved by the first and third authors, imply Theorem~$star$ of Graber and Vakil, and we provide an explicit algorithm for expressing any tautological class on $overline{mathcal{M}}_{g,n}$ of sufficiently high codimension as a boundary class.
