On the Witten-Reshetikhin-Turaev representations of mapping class groups
HTML articles powered by AMS MathViewer
- by Patrick M. Gilmer PDF
- Proc. Amer. Math. Soc. 127 (1999), 2483-2488 Request permission
Abstract:
We consider a central extension of the mapping class group of a surface with a collection of framed colored points. The Witten-Reshetikhin-Turaev TQFTs associated to $SU(2)$ and $SO(3)$ induce linear representations of this group. We show that the denominators of matrices which describe these representations over a cyclotomic field can be restricted in many cases. In this way, we give a proof of the known result that if the surface is a torus with no colored points, the representations have finite image.References
- Michael Atiyah, The geometry and physics of knots, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1990. MR 1078014, DOI 10.1017/CBO9780511623868
- C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel, Topological quantum field theories derived from the Kauffman bracket, Topology 34 (1995), no.Β 4, 883β927. MR 1362791, DOI 10.1016/0040-9383(94)00051-4
- L. Funar, TQFT representations of mapping class groups, Preprint 1997.
- Patrick M. Gilmer, Invariants for one-dimensional cohomology classes arising from TQFT, Topology Appl. 75 (1997), no.Β 3, 217β259. MR 1430088, DOI 10.1016/S0166-8641(96)00090-9
- β, Turaev-Viro Modules of Satellite Knots, Knots 96 (S. Suzuchi, ed.), World Scientific, 1997, pp. 337-363.
- Lisa C. Jeffrey, Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation, Comm. Math. Phys. 147 (1992), no.Β 3, 563β604. MR 1175494
- Hitoshi Murakami, Quantum $\textrm {SU}(2)$-invariants dominate Cassonβs $\textrm {SU}(2)$-invariant, Math. Proc. Cambridge Philos. Soc. 115 (1994), no.Β 2, 253β281. MR 1277059, DOI 10.1017/S0305004100072078
- Hitoshi Murakami, Quantum $\textrm {SO}(3)$-invariants dominate the $\textrm {SU}(2)$-invariant of Casson and Walker, Math. Proc. Cambridge Philos. Soc. 117 (1995), no.Β 2, 237β249. MR 1307078, DOI 10.1017/S0305004100073084
- G. Masbaum, J. Roberts, A simple proof of integrality of quantum invariants at prime roots of unity, Math. Proc. Camb. Phil Soc. 121 (1997), 443β454.
- G. Masbaum and J. D. Roberts, On central extensions of mapping class groups, Math. Ann. 302 (1995), no.Β 1, 131β150. MR 1329450, DOI 10.1007/BF01444490
- H. R. Morton and P. Strickland, Jones polynomial invariants for knots and satellites, Math. Proc. Cambridge Philos. Soc. 109 (1991), no.Β 1, 83β103. MR 1075123, DOI 10.1017/S0305004100069589
- Frank Quinn, Lectures on axiomatic topological quantum field theory, Geometry and quantum field theory (Park City, UT, 1991) IAS/Park City Math. Ser., vol. 1, Amer. Math. Soc., Providence, RI, 1995, pp.Β 323β453. MR 1338394, DOI 10.1090/pcms/001/05
- N. Reshetikhin and V. G. Turaev, Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no.Β 3, 547β597. MR 1091619, DOI 10.1007/BF01239527
- Pierre Samuel, Algebraic theory of numbers, Houghton Mifflin Co., Boston, Mass., 1970. Translated from the French by Allan J. Silberger. MR 0265266
- V. G. Turaev, Quantum invariants of knots and 3-manifolds, De Gruyter Studies in Mathematics, vol. 18, Walter de Gruyter & Co., Berlin, 1994. MR 1292673
- K. Walker, On Wittenβs 3-manifold invariants, preprint, 1991.
- C. T. C. Wall, Non-additivity of the signature, Invent. Math. 7 (1969), 269β274. MR 246311, DOI 10.1007/BF01404310
- Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no.Β 3, 351β399. MR 990772
- Gretchen Wright, The Reshetikhin-Turaev representation of the mapping class group, J. Knot Theory Ramifications 3 (1994), no.Β 4, 547β574. MR 1304402, DOI 10.1142/S021821659400040X
- Gretchen Wright, The Reshetikhin-Turaev representation of the mapping class group at the sixth root of unity, J. Knot Theory Ramifications 5 (1996), no.Β 5, 721β739. MR 1414096, DOI 10.1142/S0218216596000412
Additional Information
- Patrick M. Gilmer
- Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
- MR Author ID: 73695
- Email: gilmer@math.lsu.edu
- Received by editor(s): June 23, 1997
- Received by editor(s) in revised form: November 5, 1997
- Published electronically: April 15, 1999
- Additional Notes: This research was partially supported by a grant from the Louisiana Education Quality Support Fund.
- Communicated by: Ronald A. Fintushel
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 2483-2488
- MSC (1991): Primary 57M99
- DOI: https://doi.org/10.1090/S0002-9939-99-04796-6
- MathSciNet review: 1487369