Abstract
The goal of this discussion is to explain how quantum groups arise in three-dimensional topological quantum field theories (TQFTs). Of course, “explain how” is not the job of science, and perhaps you will find other explanations more satisfying. Let me explain!
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References
V. F. R. Jones. A polynomial invariant of knots via von Neumann algebras. Bull. Amer. Math. Soc., 12 (1): 103–111, 1985.
E. Witten. Quantum field theory and the Jones polynomial. Commun. Math. Phys., 121 (3): 351–399, 1989.
N. Y. Reshetikhin and V. G. Turaev. Ribbon graphs and their invariants derived from quantum groups. Commun. Math. Phys., 127 (1): 1–26, 1990.
N. Y. Reshetikhin and V. G. Turaev. Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math., 103 (3): 547–597, 1991.
V. G. Turaev. Quantum Invariants of Knots and 3-Manifolds,volume 18 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1994.
R. Dijkgraaf and E. Witten. Topological gauge theories and group cohomology. Commun. Math. Phys., 129 (2): 393–429, 1990.
R. Dijkgraaf, V. Pasquier, and P. Roche. Quasi Hopf algebras, group cohomology and orbifold models. In Recent Advances in Field Theory, (Annecy-le-Vieux, 1990), volume 18B of Nuclear Phys. B. Proc. Suppl., 1991. North-Holland, Amsterdam, pages 60–72.
D. Altschuler and A. Coste. Quasi-quantum groups, knots, 3-manifolds, and topological field theory. Commun. Math. Phys., 150 (1): 83–107, 1992.
D. S. Freed. Higher algebraic structures and quantization. Commun. Math. Phys.,159: 343–398, 1994.
D. S. Freed and F. Quinn. Chern-Simons theory with finite gauge group. Commun. Math. Phys., 156 (3): 435–472, 1993.
D. S. Freed. Extended structure in topological quantum field theory. In L. H. Kauffman and R. A. Baadhio, eds., Quantum Topology. volume 3 of Series on Knots and Everything, World Scientific, River Edge, NJ, pages 162–173, 1993.
D. S. Freed. Characteristic numbers and generalized path integrals. In Geometry, Topology, and Physics. volume VI of Conf. Proc. Lecture Notes Geom. Topology, International Press, Cambridge, MA, pages 126–138, 1995.
D. S. Freed. Lectures in topological quantum field theory. In L. A. Ibort and M. A. Rodríguez, eds., Integrable Systems, Quantum Groups, and Quantum Field Theories. Kluwer Academic Publishers, Dordrecht, pages 95–156, 1993.
D. S. Freed. Classical Chern-Simons theory. I. Ann. Math., 115 (2): 237–303, 1995.
J. Mickelsson. Kac-Moody groups and the Dirac determinant line bundle. In Topological and Geometrical Methods in Field Theory,(Espoo, 1986), 1986. World Scientific, Teaneck, NJ, pages 117–131.
S. MacLane. Categories for the Working Mathematician, volume 5 of Graduate Texts in Mathematics. Springer Verlag, New York, 1971.
E. Verlinde. Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys., B300 (3): 360–376, 1988.
M. M. Kapranov and V. A. Voevodsky. 2-categories and Zamolodchikov tetrahedra equations. In Algebraic Groups and their Generalizations: Quantum and Infinite-Dimensional Methods, (University Park, PA, 1991), volume 56, Part 2 of Proc. Sympos. Pure Math., 1994. Amer. Math. Soc., Providence, RI, pages 177–259.
R. Lawrence. Triangulations, categories and extended topological field theories. In L. H. Kauffman and R. A. Baadhio, eds., Quantum Topology. volume 3 of Series on Knots and Everything, World Scientific, Ridge River, NJ, pages 191–208, 1993.
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Freed, D.S. (1999). Quantum Groups from Path Integrals. In: Semenoff, G., Vinet, L. (eds) Particles and Fields. CRM Series in Mathematical Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1410-6_3
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DOI: https://doi.org/10.1007/978-1-4612-1410-6_3
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