482Accesses
120Citations
26 Altmetric
3Mentions
Abstract
We construct in detail a 2+1 dimensional gauge field theory with finite gauge group. In this case the path integral reduces to a finite sum, so there are no analytic problems with the quantization. The theory was originally introduced by Dijkgraaf and Witten without details. The point of working it out carefully is to focus on the algebraic structure, and particularly the construction of quantum Hilbert spaces on closed surfaces by cutting and pasting. This includes the “Verlinde formula”. The careful development may serve as a model for dealing with similar issues in more complicated cases.
This is a preview of subscription content,log in via an institution to check access.
Access this article
Subscribe and save
- Starting from 10 chapters or articles per month
- Access and download chapters and articles from more than 300k books and 2,500 journals
- Cancel anytime
Buy Now
Price includes VAT (Japan)
Instant access to the full article PDF.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, books and news in related subjects, suggested using machine learning.References
[A] Atiyah, M.F.: Topological quantum field theory. Publ. Math. Inst. Hautes Etudes Sci. (Paris)68, 175–186 (1989)
[B] Brylinski, J.-L.: private communication
[BM] Brylinski, J.-L., McLaughlin, D.A.: The geometry of degree four characteristic classes and of line bundles on loop spaces I. Preprint, 1992
[CF] Conner, P.E., Floyd, E.E.: The Relationship of Cobordism to K-Theories. Lecture Notes in Mathematics, Vol.28, Berlin, Heidelberg, New York: Springer 1966
[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)
[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.: Higher line bundles. In preparation
[F3] Freed, D.S.: Classical Chern-Simons Theory, Part 2. In preparation
[F4] Freed, D.S.: Locality and integration in topological field theory. XIX International Colloquium on Group Theoretical Methods in Physics, Anales de fisica, monografias, Ciemat (to appear)
[F5] Freed, D.S.: Higher algebraic structures and quantization. Commun. Math. Phys. (to appear)
[K] Kontsevich, M.: Rational conformal field theory and invariants of 3-dimensional manifolds. Preprint
[MM] Milnor, J., Moore, J.: On the structure of Hopf algebras. Ann. Math.81, 211–264 (1965)
[MS] Moore, G., Seiberg, N.: Lectures on RCFT, Physics, Geometry, and Topology (Banff, AB, 1989), NATO Adv. Sci. Inst. Ser. B: Phys.238, New York: Plenum 1990, pp. 263–361
[Mac] MacLane, S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, Volume5, Berlin, Heidelberg, New York: Springer 1971
[Q1] Quinn, F.: Topological foundations of topological quantum field theory. Preprint, 1991
[Q2] Quinn, F.: Lectures on axiomatic topological quantum field theory. Preprint, 1992
[S1] Segal, G.: The definition of conformal field theory. Preprint
[S2] Segal, G.: Private communication
[Se] Serre, J.-P.: Private communication
[V] Verlinde, E.: Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys.B300, 360–376 (1988)
[W] Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.121, 351–399 (1989)
[Wa] Walker, K.: On Witten's 3-manifold invariants. Preprint, 1991
[Y] 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
Department of Mathematics, University of Texas at Austin, 78712, Austin, TX, USA
Daniel S. Freed
Department of Mathematics, Virginia Polytechnical Institute, 24061, Blacksburg, VA, USA
Frank Quinn
- Daniel S. Freed
Search author on:PubMed Google Scholar
- Frank Quinn
Search author on:PubMed Google Scholar
Additional information
Communicated by A. Jaffe
The first author is supported by NSF grant DMS-8805684, an Alfred P. Sloan Research Fellowship, a Presidential Young Investigators award, and by the O'Donnell Foundation. The second author is supported by NSF grant DMS-9207973
Rights and permissions
About this article
Cite this article
Freed, D.S., Quinn, F. Chern-Simons theory with finite gauge group.Commun.Math. Phys.156, 435–472 (1993). https://doi.org/10.1007/BF02096860
Received:
Revised:
Issue date:
Share this article
Anyone you share the following link with will be able to read this content:
Sorry, a shareable link is not currently available for this article.
Provided by the Springer Nature SharedIt content-sharing initiative