Communications in Mathematical Physics

, Volume 156, Issue 3, pp 435–472 | Cite as

Chern-Simons theory with finite gauge group

  • Daniel S. Freed
  • Frank Quinn


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.


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A] Atiyah, M.F.: Topological quantum field theory. Publ. Math. Inst. Hautes Etudes Sci. (Paris)68, 175–186 (1989)Google Scholar
  2. [B] Brylinski, J.-L.: private communicationGoogle Scholar
  3. [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
  4. [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
  5. [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
  6. [DVVV] Dijkgraaf, R., Vafa, C., Verlinde, E., Verlinde, H.: Operator algebra of orbifold models. Commun. Math. Phys.123, 485–526 (1989)CrossRefGoogle Scholar
  7. [DW] Dijkgraaf, R., Witten, E.: Topological gauge theories and group cohomology. Commun. Math. Phys.129, 393–429 (1990)Google Scholar
  8. [Fg] Ferguson, K.: Link invariants associated to TQFT's with finite gauge group. Preprint, 1992Google Scholar
  9. [F1] Freed, D.S.: Classical Chern-Simons Theory, Part 1. Adv. Math. (to appear)Google Scholar
  10. [F2] Freed, D.S.: Higher line bundles. In preparationGoogle Scholar
  11. [F3] Freed, D.S.: Classical Chern-Simons Theory, Part 2. In preparationGoogle Scholar
  12. [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
  13. [F5] Freed, D.S.: Higher algebraic structures and quantization. Commun. Math. Phys. (to appear)Google Scholar
  14. [K] Kontsevich, M.: Rational conformal field theory and invariants of 3-dimensional manifolds. PreprintGoogle Scholar
  15. [MM] Milnor, J., Moore, J.: On the structure of Hopf algebras. Ann. Math.81, 211–264 (1965)Google Scholar
  16. [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
  17. [Mac] MacLane, S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, Volume5, Berlin, Heidelberg, New York: Springer 1971Google Scholar
  18. [Q1] Quinn, F.: Topological foundations of topological quantum field theory. Preprint, 1991Google Scholar
  19. [Q2] Quinn, F.: Lectures on axiomatic topological quantum field theory. Preprint, 1992Google Scholar
  20. [S1] Segal, G.: The definition of conformal field theory. PreprintGoogle Scholar
  21. [S2] Segal, G.: Private communicationGoogle Scholar
  22. [Se] Serre, J.-P.: Private communicationGoogle Scholar
  23. [V] Verlinde, E.: Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys.B300, 360–376 (1988)CrossRefGoogle Scholar
  24. [W] Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.121, 351–399 (1989)CrossRefGoogle Scholar
  25. [Wa] Walker, K.: On Witten's 3-manifold invariants. Preprint, 1991Google Scholar
  26. [Y] Yetter, D.N.: Topological quantum field theories associated to finite groups and crossedG-sets. J. Knot Theory and its Ramifications1, 1–20 (1992)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Daniel S. Freed
    • 1
  • Frank Quinn
    • 2
  1. 1.Department of MathematicsUniversity of Texas at AustinAustinUSA
  2. 2.Department of MathematicsVirginia Polytechnical InstituteBlacksburgUSA

Personalised recommendations