Topological field theories on open-closed r-spin surfaces

Stern, Walker H.; Szegedy, Lóránt

doi:10.1016/j.topol.2022.108062

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Topological field theories on open-closed r-spin surfaces

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Stern, Walker H.
Max Planck Institute for Mathematics, Max Planck Society;

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Szegedy, Lóránt
Max Planck Institute for Mathematics, Max Planck Society;

External Resource

https://doi.org/10.1016/j.topol.2022.108062
(Publisher version)

https://doi.org/10.48550/arXiv.2004.14181
(Preprint)

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Citation

Stern, W. H., & Szegedy, L. (2022). Topological field theories on open-closed r-spin surfaces. Topology and its Applications, 312: 108062. doi:10.1016/j.topol.2022.108062.

Cite as: https://hdl.handle.net/21.11116/0000-000A-55E9-0

Abstract

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 $\Lambda_r$-Frobenius algebras". We additionally extend the
state sum construction of closed $r$-spin TFTs from a $\Lambda_r$-Frobenius
algebra $A$ with invertible window element of Novak and Runkel-Szegedy to the
open-closed case. The corresponding knowledgeable $\Lambda_r$-Frobenius algebra
is $A$ together with the $\mathbb{Z}/r$-graded center of $A$.