Fully extended \(r\)-spin TQFTs (Q6149228)

Summary: 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.