Fully extended $r$-spin TQFTs

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 \mathrm{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 \mathrm{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.