论文标题
开放式$ r $ -spin表面上的拓扑字段理论
Topological field theories on open-closed $r$-spin surfaces
论文作者
论文摘要
在本文中，我们在表面上的$ r $ spin结构的两个模型之间建立了联系：诺瓦克和runkel-szegedy的标记PLCW分解，以及Dyckerhoff-Kapranov的结构化图。我们使用这些模型来描述$ r $ spin结构在开放式bordism上，从而导致二维开放式封闭$ r $ r $ r $ -spin bordism类别的发电机和关系表征。这导致根据代数结构的二维开放式封闭理论的分类，我们称我们为“知识渊博的$λ_r$ -Frobenius代数”。此外，我们还将$ r $ -sspin TFTS的状态总和从$λ_r$ -FROBENIUS代数$ a $ a $带有Novak和Runkel-Szegedy的可逆窗口元素到开放式案例。相应的知识渊博的$λ_R$ -FROBENIUS代数为$ a $，$ \ Mathbb {z}/r $ $ a $ a $。
In this article, we establish a connection between two models for $r$-spin structures on surfaces: the marked PLCW decompositions of Novak and Runkel-Szegedy, and the structured graphs of Dyckerhoff-Kapranov. We use these models to describe $r$-spin structures on open-closed bordisms, leading to a generators-and-relations characterization of the 2-dimensional open-closed $r$-spin bordism category. This results in a classification of 2-dimensional open closed field theories in terms of algebraic structures we term "knowledgeable $Λ_r$-Frobenius algebras". We additionally extend the state sum construction of closed $r$-spin TFTs from a $Λ_r$-Frobenius algebra $A$ with invertible window element of Novak and Runkel-Szegedy to the open-closed case. The corresponding knowledgeable $Λ_r$-Frobenius algebra is $A$ together with the $\mathbb{Z}/r$-graded center of $A$.
