Abstract
We derive (quasi-)quantum groups in 2+1 dimensional topological field theory directly from the classical action and the path integral. Detailed computations are carried out for the Chern-Simons theory with finite gauge group. The principles behind our computations are presumably more general. We extend the classical action in ad+1 dimensional topological theory to manifolds of dimension less thand+1. We then “construct” a generalized path integral which ind+1 dimensions reduces to the standard one and ind dimensions reproduces the quantum Hilbert space. In a 2+1 dimensional topological theory the path integral over the circle is the category of representations of a quasi-quantum group. In this paper we only consider finite theories, in which the generalized path integral reduces to a finite sum. New ideas are needed to extend beyond the finite theories treated here.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[AC] Altschuler, D., Coste, A.: Quasi-quantum groups, knots, three-manifolds, and topological field theory. Commun. Math. Phys.150, 83–107 (1992)
[A] Atiyah, M.F.: Topological quantum field theory. Publ. Math. Inst. Hautes Etudes Sci. (Paris)68, 175–186 (1989)
[Be] Benabou, J.: Introduction to bicategories. Lect. Notes in Math., vol. 47. Berlin, Heidelberg, New York: Springer 1968, pp. 1–71
[Br] Breen, L.: Théorie de Schreier supérieure. Ann. Sci. Ecole Norm. Sup. (4)25, 465–514 (1992)
[BMc] Brylinski, J.-L., McLaughlin, D.A.: The geometry of degree four characteristic classes and of line bundles on loop spaces. I. Preprint, 1992
[DM] Deligne, P., Milne, J.S.: Tannakian categories. Lect. Notes in Math., vol. 900. Berlin, Heidelberg, New York: Springer 1982, pp. 101–228
[DPR] Dijkgraaf, R., Pasquier, V., Roche, P.: Quasi-quantum groups related to orbifold models. Nucl. Phys. B. Proc. Suppl.18B, 60–72 (1990)
[DVVV] Dijkgraaf, R., Vafa, C., Verlinde, E., Verlinde, H.: Operator algebra of orbifold models. Commun. Math. Phys.123, 485–526 (1989)
[DW] Dijkgraaf, R., Witten, E.: Topological gauge theories and group cohomology. Commun. Math. Phys.129, 393–429 (1990)
[Dr] Drinfeld, V.G.: Quasi-Hopf algebras and Knizhnik-Zamolodchikov equations. Problems of modern quantum field theory, Alushta, 1989, Belavin, A. et al. (eds.) Berlin, Heidelberg, New York: Springer 1989, pp. 1–13
[Fg] Ferguson, K.: Link invariants associated to TQFT's with finite gauge group. Preprint, 1992
[F1] Freed, D.S.: Classical Chern-Simons theory, Part 1. Adv. Math., to appear
[F2] Freed, D.S.: Classical Chern-Simons theory, Part 2. In preparation
[F3] Freed, D.S.: Extended structures in topological quantum field theory. Preprint, 1993
[FG] Freed, D.S., Quinn, F.: Chern-Simons theory with finite gauge group. Commun. Math. Phys., to appear
[Gi] Giraud, J.: Cohomologie non-abélienne. Ergeb. der Math., vol. 64, Berlin, Heidelberg, New York: Springer 1971
[JS] Joyal, A., Street, R.: The geometry of tensor calculus. I. Adv. Math.88, 55–112 (1991)
[KV] Kapranov, M.M., Voevodsky, V.A.: 2-Categories and Zamolodchikov tetrahedra equations. Preprint, 1992
[KR] Kazhdan, D., Reshetikhin, N.Y.: In preparation
[Ko] Kontsevich, M.: Rational conformal field theory and invariants of 3-dimensional manifolds. Preprint
[L] Lawrence, R.: In preparation
[Mac] MacLane, S.: Categories for the working mathematician. Graduate Texts in Mathematics, Vol. 5. Berlin, Heidelberg, New York: Springer 1971
[Ma1] Majid, S.: Tannaka-Krein theorem for quasi-Hopf algebras and other results. Deformation Theory and Quantum Groups with Applications to Mathematical Physics, Contemporary Mathematics, vol. 134. Providence, RI: Am. Math. Soc., 1992, pp. 219–232
[Ma2] Majid, S.: Braided groups. Preprint, 1990
[Ma3] Majid, S.: Quasi-quantum groups as internal symmetries of topological quantum field theories. Lett. Math. Phys.22, 83–90 (1991)
[MS] Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys.123, 177–254 (1989)
[Q1] Quinn, F.: Lectures on axiomatic topological quantum field theory. Preprint, 1992
[Q2] Quinn, F.: Topological foundations of topological quantum field theory. Preprint 1991
[RT] Reshetikhin, N.Y., Turaev, V.G.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math.103, 547–597 (1991)
[S1] Segal, G.: The definition of conformal field theory. Preprint
[S2] Segal, G.: Private communication
[Sh] Shum: Tortile tensor categories. J. Pure Appl. Alg., to appear
[V] Verlinde, E.: Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys. B300, 360–376 (1998)
[W] Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.121, 351–399 (1989)
[Y1] Yetter, D.N.: Topology '90. Apanasov, B., Neumann, W.D., Reid, A.W., Siebenmann, L. (eds.) De Gruyter, 1992, pp. 399–444
[Y2] Yetter, D.N.: State-sum invariants of 3-manifolds associated to artinian semisimple tortile categories. Preprint
[Y3] Yetter, D.N.: Topological quantum field theories associated to finite groups and crossedG-sets. J. Knot Theory and its Ramifications1, 1–20 (1992)
Author information
Authors and Affiliations
Additional information
Communicated by G. Felder
The author is supported by NSF grant DMS-8805684, a Presidential Young Investigators award DMS-9057144, and by the O'Donnell Foundation. He warmly thanks the Geometry Center at the University of Minnesota for their hospitality while this work was undertaken
Rights and permissions
About this article
Cite this article
Freed, D.S. Higher algebraic structures and quantization. Commun.Math. Phys. 159, 343–398 (1994). https://doi.org/10.1007/BF02102643
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02102643