Topological field theory on r-spin surfaces and the Arf-invariant

Runkel I, Szegedy L. 2021. Topological field theory on r-spin surfaces and the Arf-invariant. Journal of Mathematical Physics. 62(10), 102302.


Journal Article | Published | English

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Author
Runkel, Ingo; Szegedy, LorantISTA
Department
Abstract
We give a combinatorial model for r-spin surfaces with parameterized boundary based on Novak (“Lattice topological field theories in two dimensions,” Ph.D. thesis, Universität Hamburg, 2015). The r-spin structure is encoded in terms of ℤ𝑟-valued indices assigned to the edges of a polygonal decomposition. This combinatorial model is designed for our state-sum construction of two-dimensional topological field theories on r-spin surfaces. We show that an example of such a topological field theory computes the Arf-invariant of an r-spin surface as introduced by Randal-Williams [J. Topol. 7, 155 (2014)] and Geiges et al. [Osaka J. Math. 49, 449 (2012)]. This implies, in particular, that the r-spin Arf-invariant is constant on orbits of the mapping class group, providing an alternative proof of that fact.
Publishing Year
Date Published
2021-10-01
Journal Title
Journal of Mathematical Physics
Acknowledgement
We would like to thank Nils Carqueville, Tobias Dyckerhoff, Jan Hesse, Ehud Meir, Sebastian Novak, Louis-Hadrien Robert, Nick Salter, Walker Stern, and Lukas Woike for helpful discussions and comments. L.S. was supported by the DFG Research Training Group 1670 “Mathematics Inspired by String Theory and Quantum Field Theory.”
Volume
62
Issue
10
Article Number
102302
ISSN
IST-REx-ID

Cite this

Runkel I, Szegedy L. Topological field theory on r-spin surfaces and the Arf-invariant. Journal of Mathematical Physics. 2021;62(10). doi:10.1063/5.0037826
Runkel, I., & Szegedy, L. (2021). Topological field theory on r-spin surfaces and the Arf-invariant. Journal of Mathematical Physics. AIP. https://doi.org/10.1063/5.0037826
Runkel, Ingo, and Lorant Szegedy. “Topological Field Theory on R-Spin Surfaces and the Arf-Invariant.” Journal of Mathematical Physics. AIP, 2021. https://doi.org/10.1063/5.0037826.
I. Runkel and L. Szegedy, “Topological field theory on r-spin surfaces and the Arf-invariant,” Journal of Mathematical Physics, vol. 62, no. 10. AIP, 2021.
Runkel I, Szegedy L. 2021. Topological field theory on r-spin surfaces and the Arf-invariant. Journal of Mathematical Physics. 62(10), 102302.
Runkel, Ingo, and Lorant Szegedy. “Topological Field Theory on R-Spin Surfaces and the Arf-Invariant.” Journal of Mathematical Physics, vol. 62, no. 10, 102302, AIP, 2021, doi:10.1063/5.0037826.
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