Open $r$-spin theory in genus one, and the Gelfand-Dikii wave function

Abstract

We construct the $g=1$ sector of the open $r$-spin theory, that is, an open $r$-spin theory on the moduli space of cylinders. This is the second construction of a $g>0$ open intersection theory, which includes descendents (the first is the all genus construction of the intersection theory on moduli of open Riemann surfaces with boundaries [23,30], whose $g=1$ case equals to the $r=2$ case of our construction). Unlike the construction of [30], in order to construct the $r$-spin cylinder theory we had to overcome the foundational problem of dimension jump loci, which in analogous closed theories has been treated using virtual fundamental class techniques, that are currently absent in the open setting. For this reason our construction is much more involved, and relies on the point insertion technique developed in [31,32]. We prove that the open $g=1$ potential equals, after a coordinate change, to the $g=1$ part of the Gelfand-Dikii wave function, thus confirming a conjecture of [7]. We also prove that our $g=1$ intersection numbers satisfy a $g=1$ recursion, also predicted in [7,15]. This recursion is the $g=1$ analogue of Solomon's famous $g=0$ Open WDVV equation [25], with descendents, and is also the universal $g=1$ recursion for $F$-Cohomological field theories [1]. Again, this is first geometric construction which is not the $g=1$ sector of [23,30], proven to satisfy this universal recursion.

Figures (6)

• Figure 1: The five types of nodes on a nodal marked cylinder.
• Figure 2: The thicker lines representing the intervals $I_h$ and $I_{h'}$ associated to the half-nodes $n_h$ and $n_{h'}$ are drawn over the thinner boundary lines. The image on the right represents a point in the moduli space that is close to the image on the left, where the node is smoothed.
• 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 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 markings coming from the point insertion procedure is represented by $*$; the dashed line between the new markings indicates that they come from the same node.
• Figure 4: An example of two $(r,0)$-disks lying on two boundaries $\text{bd}_{BI}$ and $\text{bd}_{AI}$ paired by $PI$. The component $C_1^{BI}$ of the $(r,0)$-disk on the left has a type-BI node, while the component $D_1^{AI}$ of the $(r,0)$-disk on the left has a type-AI node. The shaded irreducible component only contains a legal half-node (corresponding to a type-AI node) and an internal tail in $I^{PI}$.
• Figure 5: An example of two $(r,0)$-cylinders paired by $\text{PI}$, where the left one has a non-separating BI-type boundary node, and the right one has an separating AI-type boundary node, and a dashed line connecting a boundary tail and a internal tail on the same connected component.

Theorems & Definitions (83)

• Theorem 1.1
• Conjecture 1: The rGD wave function conjecture of BCT3
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 2.1
• Remark 2.2
• Remark 2.4