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Quantum \(SU(2)\) faithfully detects mapping class groups modulo center. (English) Zbl 1037.57024

A Topological Quantum Field Theory contains projective representations of mapping class groups. This paper establishes an asymptotic faithfulness result for the representations arising from the \(SU(2)\) family of TQFTs. This family is indexed by an integer \(r\). The exact statement says that a non-central element of the mapping class group acts projectively nontrivially for \(r\) big enough. This paper is based on the Kauffman bracket construction of the \(SU(2)\) TQFTs, together with a (somewhat tricky) topological lemma. An asymptotic faithfulness result using geometric quantization has been obtained by J. Andersen [arXiv:math. QA/0204084].

MSC:

57R56 Topological quantum field theories (aspects of differential topology)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
22E46 Semisimple Lie groups and their representations
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
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