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

TL;DR

This work extends open $r$-spin theories to graded $r$-spin surfaces with multiple boundary and internal twists by constructing moduli spaces, the open Witten bundle, and a canonical relative orientation framework. It develops a robust orientation theory for genus $0$ disks and genus $1$ cylinders, including how these orientations behave under boundary gluing and the point insertion operation, which identifies boundary strata with internal insertions. A key technical advance is handling genus $1$ Witten bundles by removing the dimension-jump locus and using gluing along boundary strata to produce canonically oriented bundles on glued moduli spaces. The paper then introduces point insertion and reduced $(r,rak{h})$-surfaces, along with $(r,rak{h})$-graphs, to implement a general gluing strategy that yields well-defined open intersection theories; sequels TZ2 and TZ3 promise to construct intersection theories indexed by ${rak h}$ and connect genus-one potentials to the Gelfand-Dikii hierarchy. Together, these results lay a foundation for open $r$-spin intersection theory, its enumerative applications, and links to mirror symmetry and integrable systems.

Abstract

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].

Figures (7)

• Figure 1: The five types of nodes on a nodal marked surface.
• Figure 2: Dual graphs corresponding to the surfaces in Figure \ref{['fig node type']}. We represent by thick rectangles the open vertices and by thick circles the open vertices. The boundary (half-)edges are represented by ordinary lines, the internal (half-)edges are represented by double lines, and the contracted boundary tails are represented by segments with black endpoints.
• Figure 3: In point insertion procedure we glue ${\overline{\mathcal{M}}}_1$ and ${\overline{\mathcal{M}}}_2$ together along their isomorphic boundaries $\text{bd}_{BI}$ and $\text{bd}_{AI}$. The first isomorphism follows from the decomposition property for the boundary NS nodes; the second isomorphism holds because the moduli ${\overline{\mathcal{M}}}^{1/r}_{0.\{r-2-2h\},\{h\}}$ (represented by the smallest bubble in the figure) is a single point. The new marked points coming from the point insertion procedure are represented by $*$; the dashed line between the new marked points indicates that they come from the same node.
• Figure 4: An example of gluing $1$-dimensional moduli spaces by point insertion
• Figure 5: An example of gluing $2$-dimensional moduli spaces by point insertion. We omit some twists of markings because they are all equal to $0$.

Theorems & Definitions (75)

• Remark 2.1
• Remark 2.3
• Proposition 2.4
• proof
• Definition 2.5
• Definition 2.6
• Remark 2.7
• Theorem 2.8
• proof
• Remark 2.9