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 [Nov]. The r-spin structure is encoded in terms of 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 [Ran, GG].We use the combinatorial model to give a 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 [Ran, GG]. 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.

Fully extended $\boldsymbol{r}$-spin TQFTs

“…A 1-spin structure is the same as an orientation, and a 2-spin structure is usually called a spin structure. Moreover, we can identify framings with 0-spin structures by noting that the fibres of a 0-spin bundle are contractible, see [RS,Prop. 2.2].…”

“…There is a unique lift Γ : fixing it at one point, as the fibres are discrete. We define δ(c) ∈ Z r to be the holonomy of Γ, which only depends on c; for more details of this construction we refer to [RW] or [RS,Sect. 5.2].…”

Spin Structures and Exact Dualities in Low Dimensions

“…must be chosen so that, for a fixed c( ) in (43), the above product agrees in sign with the modified Gauss law G ∨ f = 1 based on eq. ( 42).…”

Fermionization and boundary states in 1+1 dimensions

“…up to a subtle phase which does not concern us in this paper. 15 Here, Ŝij describes the action of S ∈ SL(2, Z) on the space of torus conformal blocks with an insertion of the simple current υ.…”

“…It would be interesting to revisit and develop these points further. See e.g [15][16][17]. for recent works 2.…”