# Chern-Simons theory with finite gauge group

Article

- 235 Downloads
- 51 Citations

## 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.

## Keywords

Neural Network Statistical Physic Hilbert Space Field Theory Complex System
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- [A] Atiyah, M.F.: Topological quantum field theory. Publ. Math. Inst. Hautes Etudes Sci. (Paris)
**68**, 175–186 (1989)Google Scholar - [B] Brylinski, J.-L.: private communicationGoogle Scholar
- [BM] Brylinski, J.-L., McLaughlin, D.A.: The geometry of degree four characteristic classes and of line bundles on loop spaces I. Preprint, 1992Google Scholar
- [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 1966Google Scholar - [DPR] Dijkgraaf, R., Pasquier, V., Roche, P.: Quasi-quantum groups related to orbifold models. Nucl. Phys. B. Proc. Suppl.
**18B**, 60–72 (1990)Google Scholar - [DVVV] Dijkgraaf, R., Vafa, C., Verlinde, E., Verlinde, H.: Operator algebra of orbifold models. Commun. Math. Phys.
**123**, 485–526 (1989)CrossRefGoogle Scholar - [DW] Dijkgraaf, R., Witten, E.: Topological gauge theories and group cohomology. Commun. Math. Phys.
**129**, 393–429 (1990)Google Scholar - [Fg] Ferguson, K.: Link invariants associated to TQFT's with finite gauge group. Preprint, 1992Google Scholar
- [F1] Freed, D.S.: Classical Chern-Simons Theory, Part 1. Adv. Math. (to appear)Google Scholar
- [F2] Freed, D.S.: Higher line bundles. In preparationGoogle Scholar
- [F3] Freed, D.S.: Classical Chern-Simons Theory, Part 2. In preparationGoogle Scholar
- [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)Google Scholar
- [F5] Freed, D.S.: Higher algebraic structures and quantization. Commun. Math. Phys. (to appear)Google Scholar
- [K] Kontsevich, M.: Rational conformal field theory and invariants of 3-dimensional manifolds. PreprintGoogle Scholar
- [MM] Milnor, J., Moore, J.: On the structure of Hopf algebras. Ann. Math.
**81**, 211–264 (1965)Google Scholar - [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. 263–361Google Scholar - [Mac] MacLane, S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, Volume
**5**, Berlin, Heidelberg, New York: Springer 1971Google Scholar - [Q1] Quinn, F.: Topological foundations of topological quantum field theory. Preprint, 1991Google Scholar
- [Q2] Quinn, F.: Lectures on axiomatic topological quantum field theory. Preprint, 1992Google Scholar
- [S1] Segal, G.: The definition of conformal field theory. PreprintGoogle Scholar
- [S2] Segal, G.: Private communicationGoogle Scholar
- [Se] Serre, J.-P.: Private communicationGoogle Scholar
- [V] Verlinde, E.: Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys.
**B300**, 360–376 (1988)CrossRefGoogle Scholar - [W] Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.
**121**, 351–399 (1989)CrossRefGoogle Scholar - [Wa] Walker, K.: On Witten's 3-manifold invariants. Preprint, 1991Google Scholar
- [Y] Yetter, D.N.: Topological quantum field theories associated to finite groups and crossed
*G*-sets. J. Knot Theory and its Ramifications**1**, 1–20 (1992)CrossRefGoogle Scholar

## Copyright information

© Springer-Verlag 1993