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

  title={Topological field theory on r-spin surfaces and the Arf-invariant},
  author={Ingo Runkel and L'or'ant Szegedy},
  journal={Journal of Mathematical Physics},
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 $\mathbb{Z}_r$-valued indices assigned to the edges of a polygonal decomposition. With the help of this model we count the number of mapping class group orbits on r-spin surfaces with parametrised boundary and fixed r-spin structure on each boundary component, extending (and giving a different proof of) results in Geiges, Gonzalo (2012) and Randal… 
3 Citations
String-net models for nonspherical pivotal fusion categories
A string-net model associates a vector space to a surface in terms of graphs decorated by objects and morphisms of a pivotal fusion category modulo local relations. String-net models are usually
Area-Dependent Quantum Field Theory
Area-dependent quantum field theory is a modification of two-dimensional topological quantum field theory, where one equips each connected component of a bordism with a positive real


State-sum construction of two-dimensional functorial field theories
In this thesis we study two classes of 2-dimensional functorial field theories and give a state-sum construction of these theories. In the first part of this thesis we look at topological field
Spin Hurwitz numbers and topological quantum field theory
Spin Hurwitz numbers count ramified covers of a spin surface, weighted by the size of their automorphism group (like ordinary Hurwitz numbers), but signed $\pm 1$ according to the parity of the
Generalised spin structures on 2-dimensional orbifolds
Generalised spin structures, or r-spin structures, on a 2-dimensional orbifold \Sigma are r-fold fibrewise connected coverings (also called r-th roots) of its unit tangent bundle ST\Sigma. We
State sum construction of two-dimensional topological quantum field theories on spin surfaces
We provide a combinatorial model for spin surfaces. Given a triangulation of an oriented surface, a spin structure is encoded by assigning to each triangle a preferred edge, and to each edge an
This article treats various aspects of the geometry of the moduli of r-spin curves and its compactification . Generalized spin curves, or r-spin curves, are a natural generalization of 2-spin curves
Homology of the moduli spaces and mapping class groups of framed, r‐Spin and Pin surfaces
We give definitions of moduli spaces of framed, r‐Spin and Pin ± surfaces. We apply earlier work of the author to show that each of these moduli spaces exhibits homological stability, and we identify
Lattice topological field theory in two dimensions
The lattice definition of a two-dimensional topological field theory (TFT) is given generically, and the exact solution is obtained explicitly. In particular, the set of all lattice topological field
We present a state sum construction of two-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from
The mapping class group orbits in the framings of compact surfaces
We compute the mapping class group orbits in the homotopy set of framings of a compact connected oriented surface with non-empty boundary. In the case $g > 1$ the computation is some modification of