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S. Mac Lane, Categories for the Working Mathematician
1971
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M. Atiyah, Topological Quantum Field Theories
1988
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N. Reshetikhin, V. Turaev, Invariants of 3-Manifolds via Link Polynomials and Quantum Groups
1991
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S. Majid, Braided Groups
1993
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D. Radford, The Trace Function and Hopf Algebras
1994
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V. Turaev, Quantum Invariants of Knots and 3-Manifolds
1994
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C. Blanchet, N. Habegger, G. Masbaum, P. Vogel, Topological Quantum Field Theories Derived from the Kauffman Bracket
1995
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V. Lyubashenko, Invariants of 3-Manifolds and Projective Representations of Mapping Class Groups via Quantum Groups at Roots of Unity
1995
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V. Lyubashenko, Modular Transformations for Tensor Categories
1995
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T. Ohtsuki, Invariants of 3-Manifolds Derived from Universal Invariants of Framed Links
1995
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M. Hennings, Invariants of Links and 3-Manifolds Obtained from Hopf Algebras
1996
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R. Kashaev, The Hyperbolic Volume of Knots from the Quantum Dilogarithm
1997
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T. Kerler, Genealogy of Nonperturbative Quantum-Invariants of 3-Manifolds: The Surgical Family
1997
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T. Kerler, On the Connectivity of Cobordisms and Half-Projective TQFT’s
1998
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T. Kerler, V. Lyubashenko, Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners
2001
Earlier work this paper cites.

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2019

3-Dimensional TQFTs From Non-Semisimple Modular Categories

De Renzi, Marco, Gainutdinov, Azat M., Geer, Nathan et al.

Understand

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.

Built on

S. Mac Lane, Categories for the Working Mathematician
1971
Earlier work this paper cites.
M. Atiyah, Topological Quantum Field Theories
1988
Earlier work this paper cites.
N. Reshetikhin, V. Turaev, Invariants of 3-Manifolds via Link Polynomials and Quantum Groups
1991
Earlier work this paper cites.
S. Majid, Braided Groups
1993
Earlier work this paper cites.
D. Radford, The Trace Function and Hopf Algebras
1994
Earlier work this paper cites.
V. Turaev, Quantum Invariants of Knots and 3-Manifolds
1994
Earlier work this paper cites.
C. Blanchet, N. Habegger, G. Masbaum, P. Vogel, Topological Quantum Field Theories Derived from the Kauffman Bracket
1995
Earlier work this paper cites.
V. Lyubashenko, Invariants of 3-Manifolds and Projective Representations of Mapping Class Groups via Quantum Groups at Roots of Unity
1995
Earlier work this paper cites.
V. Lyubashenko, Modular Transformations for Tensor Categories
1995
Earlier work this paper cites.
T. Ohtsuki, Invariants of 3-Manifolds Derived from Universal Invariants of Framed Links
1995
Earlier work this paper cites.
M. Hennings, Invariants of Links and 3-Manifolds Obtained from Hopf Algebras
1996
Earlier work this paper cites.
R. Kashaev, The Hyperbolic Volume of Knots from the Quantum Dilogarithm
1997
Earlier work this paper cites.
T. Kerler, Genealogy of Nonperturbative Quantum-Invariants of 3-Manifolds: The Surgical Family
1997
Earlier work this paper cites.
T. Kerler, On the Connectivity of Cobordisms and Half-Projective TQFT’s
1998
Earlier work this paper cites.
T. Kerler, V. Lyubashenko, Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners
2001
Earlier work this paper cites.