Geometry & Topology

Quantum $SU(2)$ faithfully detects mapping class groups modulo center

Michael H Freedman, Kevin Walker, and Zhenghan Wang

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The Jones–Witten theory gives rise to representations of the (extended) mapping class group of any closed surface Y indexed by a semi-simple Lie group G and a level k. In the case G=SU(2) these representations (denoted VA(Y)) have a particularly simple description in terms of the Kauffman skein modules with parameter A a primitive 4rth root of unity (r=k+2). In each of these representations (as well as the general G case), Dehn twists act as transformations of finite order, so none represents the mapping class group (Y) faithfully. However, taken together, the quantum SU(2) representations are faithful on non-central elements of (Y). (Note that (Y) has non-trivial center only if Y is a sphere with 0,1, or 2 punctures, a torus with 0,1, or 2 punctures, or the closed surface of genus =2.) Specifically, for a non-central h(Y) there is an r0(h) such that if rr0(h) and A is a primitive 4rth root of unity then h acts projectively nontrivially on VA(Y). Jones’ original representation ρn of the braid groups Bn, sometimes called the generic q–analog–SU(2)–representation, is not known to be faithful. However, we show that any braid hidBn admits a cabling c=c1,,cn so that ρN(c(h))id, N=c1++cn.

Article information

Geom. Topol., Volume 6, Number 2 (2002), 523-539.

Received: 14 September 2002
Accepted: 19 November 2002
First available in Project Euclid: 21 December 2017

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Zentralblatt MATH identifier

Primary: 57R56: Topological quantum field theories 57M27: Invariants of knots and 3-manifolds
Secondary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45] 22E46: Semisimple Lie groups and their representations 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35]

quantum invariants Jones–Witten theory mapping class groups


Freedman, Michael H; Walker, Kevin; Wang, Zhenghan. Quantum $SU(2)$ faithfully detects mapping class groups modulo center. Geom. Topol. 6 (2002), no. 2, 523--539. doi:10.2140/gt.2002.6.523.

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