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Non-semisimple extended topological quantum field theories
About this Title
Marco De Renzi
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 277, Number 1364
ISBNs: 978-1-4704-5269-8 (print); 978-1-4704-7096-8 (online)
DOI: https://doi.org/10.1090/memo/1364
Published electronically: April 1, 2022
Keywords: Extended topological quantum field theories,
non-semisimple categories,
quantum invariants,
2-categories
Table of Contents
Chapters
- Preface
- 1. Relative modular categories
- 2. Admissible cobordisms
- 3. Extension of Costantino-Geer-Patureau invariants
- 4. Combinatorial and topological properties
- 5. Graded extensions
- 6. Symmetric monoidality
- 7. Characterization of the image
- A. Unrolled quantum groups
- B. Manifolds and cobordisms with corners
- C. Signature defects
- D. Symmetric monoidal 2-categories
- E. Complete linear and graded linear categories
Abstract
We develop the general theory for the construction of Extended Topological Quantum Field Theories (ETQFTs) associated with the Costantino-Geer-Patureau quantum invariants of closed 3-manifolds. In order to do so, we introduce relative modular categories, a class of ribbon categories which are modeled on representations of unrolled quantum groups, and which can be thought of as a non-semisimple analogue to modular categories. Our approach exploits a 2-categorical version of the universal construction introduced by Blanchet, Habegger, Masbaum, and Vogel. The 1+1+1-EQFTs thus obtained are realized by symmetric monoidal 2-functors which are defined over non-rigid 2-categories of admissible cobordisms decorated with colored ribbon graphs and cohomology classes, and which take values in 2-categories of complete graded linear categories. In particular, our construction extends the family of graded 2+1-TQFTs defined for the unrolled version of quantum $\mathfrak {sl}_2$ by Blanchet, Costantino, Geer, and Patureau to a new family of graded ETQFTs. The non-semisimplicity of the theory is witnessed by the presence of non-semisimple graded linear categories associated with critical 1-manifolds.- B. Bartlett, C. Douglas, C. Schommer-Pries, J. Vicary, Modular Categories as Representations of the 3-Dimensional Bordism 2-Category, arXiv:1509.06811
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