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

article Topological field theory on r-spin surfaces and the Arf-invariant published yes Ingo Runkel author Lorant Szegedy author 7943226E-220E-11EA-94C7-D59F3DDC885E0000-0003-2834-5054 MiLe department 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. AIP Publishing2021 eng Journal of Mathematical Physics 0022-2488 1802.09978 00075563850001010.1063/5.0037826 6210 Runkel I, Szegedy L. Topological field theory on r-spin surfaces and the Arf-invariant. Journal of Mathematical Physics. 2021;62(10). doi:<a href="https://doi.org/10.1063/5.0037826">10.1063/5.0037826</a> Runkel I, Szegedy L. 2021. Topological field theory on r-spin surfaces and the Arf-invariant. Journal of Mathematical Physics. 62(10), 102302. 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 Publishing, 2021. I. Runkel, L. Szegedy, Journal of Mathematical Physics 62 (2021). Runkel, Ingo, and Lorant Szegedy. “Topological Field Theory on R-Spin Surfaces and the Arf-Invariant.” Journal of Mathematical Physics. AIP Publishing, 2021. <a href="https://doi.org/10.1063/5.0037826">https://doi.org/10.1063/5.0037826</a>. Runkel, I., & Szegedy, L. (2021). Topological field theory on r-spin surfaces and the Arf-invariant. Journal of Mathematical Physics. AIP Publishing. <a href="https://doi.org/10.1063/5.0037826">https://doi.org/10.1063/5.0037826</a> 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 Publishing, 2021, doi:<a href="https://doi.org/10.1063/5.0037826">10.1063/5.0037826</a>. 101762021-10-24T22:01:32Z2025-07-10T11:49:44Z