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

Abstract

We prove the $r$-spin cobordism hypothesis in the setting of (weak) 2-categories for every positive integer $r$: The 2-groupoid of 2-dimensional fully extended $r$-spin TQFTs with given target is equivalent to the homotopy fixed points of an induced $\textrm{Spin}_2^r$-action. In particular, such TQFTs are classified by fully dualisable objects together with a trivialisation of the $r$-th power of their Serre automorphisms. For $r=1$ we recover the oriented case (on which our proof builds), while ordinary spin structures correspond to $r=2$. To construct examples, we explicitly describe $\textrm{Spin}_2^r$-homotopy fixed points in the equivariant completion of any symmetric monoidal 2-category. We also show that every object in a 2-category of Landau--Ginzburg models gives rise to fully extended spin TQFTs, and that half of these do not factor through the oriented bordism 2-category.