Paper detail

The point insertion technique and open $r$-spin theories I: moduli and orientation

The papers [3,1,4,10] constructed an intersection theory on the moduli space of $r$-spin disks, and proved it satisfies mirror symmetry and relations with integrable hierarchies. That theory considered only disks with a single boundary state. In this work, we initiate the study of more general $r$-spin surfaces. We define graded $r$-spin surfaces with multiple internal and boundary states, together with their moduli spaces. In genus zero, the disk case, we define the associated open Witten bundle and prove that it is canonically oriented relative to the moduli space. We also describe a gluing construction for moduli spaces along boundaries, show that it lifts to the Witten bundle and relative cotangent line bundles, and that the result remains canonically relatively oriented. We then study the genus-one cylinder case. Here foundational difficulties arise because the Witten "bundle" is no longer an orbifold vector bundle. We resolve this by removing strata with incorrect fibre dimension, obtaining an orbibundle on the complement. The gluing method extends to genus one, and we prove that the Witten bundle again admits a canonical relative orientation. In the sequel [20], we construct a family of $\lfloor r/2\rfloor$ intersection theories in genus-zero indexed by $\mathfrak h\in\{0,\ldots,\lfloor r/2\rfloor-1\}$, where the $\mathfrak h$-th theory has $\mathfrak h+1$ boundary states, and compute their intersection numbers. The case $\mathfrak h=0$ recovers the theory of [3,1]. In the sequel [21], restricting to the $\mathfrak h=0$ case, we construct an intersection theory on the moduli space of $r$-spin cylinders and show that its potential yields, after a change of variables, the genus-one part of the $r$th Gelfand-Dikii wave function, proving the genus-one case of the main conjecture of [4].