3-Dimensional TQFTs From Non-Semisimple Modular Categories
Marco De Renzi, Azat M. Gainutdinov, Nathan Geer, Bertrand, Patureau-Mirand, and Ingo Runkel
TL;DR
This paper develops new 3-dimensional topological quantum field theories using non-semisimple modular categories, extending previous invariants and representations to include important quasi-Hopf algebra examples.
Contribution
It introduces a renormalization method for Lyubashenko's invariants, producing new TQFTs that incorporate non-semisimple categories previously excluded.
Findings
Constructed new 3D TQFTs from non-semisimple categories
Extended Lyubashenko's mapping class group representations
Included examples from quasi-Hopf algebra representation theory
Abstract
We use modified traces to renormalize Lyubashenko's closed 3-manifold invariants coming from twist non-degenerate finite unimodular ribbon categories. Our construction produces new topological invariants which we upgrade to 2+1-TQFTs under the additional assumption of factorizability. The resulting functors provide monoidal extensions of Lyubashenko's mapping class group representations, as discussed in arXiv:2010.14852. This general framework encompasses important examples of non-semisimple modular categories coming from the representation theory of quasi-Hopf algebras, which were left out of previous non-semisimple TQFT constructions.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Operator Algebra Research · Homotopy and Cohomology in Algebraic Topology