Topological field theory on r-spin surfaces and the Arf invariant
Abstract
We give a combinatorial model for r-spin surfaces with parametrised boundary based on Novak (2015). The r-spin structure is encoded in terms of -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 in Geiges, Gonzalo (2012) and Randal-Williams (2014). 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.
Keywords
Cite
@article{arxiv.1802.09978,
title = {Topological field theory on r-spin surfaces and the Arf invariant},
author = {Ingo Runkel and Lóránt Szegedy},
journal= {arXiv preprint arXiv:1802.09978},
year = {2024}
}
Comments
v2: 52 pages, removed classification of mapping class group orbits as suggested by referee, version to appear in Journal of Mathematical Physics