English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The papers [3, 1, 4, 10] constructed an intersection theory on the modulispace of $r$-spin disks, and proved it satisfies mirror symmetry and relationswith integrable hierarchies. That theory considered only disks with a singletype of boundary state. In this work, we initiate the study of more general$r$-spin surfaces. We define the notion of graded $r$-spin surfaces withmultiple internal and boundary states, and their moduli spaces. In $g=0,$ thedisk case, we also define the associated open Witten bundle, and prove that theWitten bundle is canonically oriented relative to the moduli space. Moreover,we describe a method for gluing several moduli spaces along certain boundaries,show that gluing lifts to the Witten bundle and relative cotangent linebundles, and that the result is again canonically relatively oriented.  In the sequel [20], we construct, based on the work of this paper, a familyof $\lfloor r/2\rfloor$ intersection theories indexed by$\mathfrak{h}\in\{0,\ldots,\lfloor r/2\rfloor-1\},$ where the $\mathfrak{h}$-thone has $\mathfrak{h}+1$ possible boundary states, and calculate theirintersection numbers. The $\mathfrak{h}=0$ theory is equivalent to the oneconstructed in [3, 1].

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