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Fully extended $\boldsymbol{r}$-spin TQFTs
- Source :
- Quantum Topol. 14 (2023), no. 3, pp. 467-532
- Publication Year :
- 2021
-
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.<br />Comment: 64 pages; v2: minor changes, Proposition 3.2 assumes a pivotal structure
Details
- Database :
- arXiv
- Journal :
- Quantum Topol. 14 (2023), no. 3, pp. 467-532
- Publication Type :
- Report
- Accession number :
- edsarx.2107.02046
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.4171/QT/193