Communications in Mathematical Physics

, Volume 141, Issue 1, pp 153–209 | Cite as

On quantum gauge theories in two dimensions

  • Edward Witten


Two dimensional quantum Yang-Mills theory is studied from three points of view: (i) by standard physical methods; (ii) by relating it to the largek limit of three dimensional Chern-Simons theory and two dimensional conformal field theory; (iii) by relating its weak coupling limit to the theory of Reidemeister-Ray-Singer torsion. The results obtained from the three points of view agree and give formulas for the volumes of the moduli spaces of representations of fundamental groups of two dimensional surfaces.


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  1. 1.
    Atiyah, M.F., Bott, R.: The Yang-Mills equations over Riemann surfaces. Phil. Trans. R. Soc. London A308, 523 (1982)Google Scholar
  2. 2.
    Goldman, W.: The symplectic nature of fundamental groups of surfaces. Adv. Math.54, 200 (1984)Google Scholar
  3. 3.
    Ray, D., Singer, I.:R-torsion and the Laplacian on Riemannian manifolds. Adv. Math.7, 145 (1971)Google Scholar
  4. 4.
    Narasimhan, M.S., Seshadri, C.: Stable and unitary bundles on a compact Riemann surface. Ann. Math.82, 540 (1965)Google Scholar
  5. 5.
    Verlinde, E.: Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys. B300, 360 (1988)Google Scholar
  6. 6.
    Tsuchiya, A., Ueno, K., Yamada, Y.: Conformal field theory on universal family of stable curves with gauge symmetries. Max-Planck-Institut preprint MPI/89/18Google Scholar
  7. 7.
    Segal, G.: Two dimensional conformal field theories and modular functors. In: IXth International Conference on Mathematical Physics (Swansea, July, 1988) Simon, B., Truman, A., Davies, I.M. (eds.). Bristol: Adam Hilger (1989) 22, and preprint (to appear)Google Scholar
  8. 8.
    Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.121, 351 (1989)Google Scholar
  9. 9.
    Apostol, T.M.: Introduction to analytic number theory. Berlin, Heidelberg, New York: Springer 1984Google Scholar
  10. 10.
    Thaddeus, M.: Conformal field theory and the moduli space of stable bundles. Oxford University preprintGoogle Scholar
  11. 11.
    Milnor, J.: Whitehead torsion. Bull. Am. Math. Soc.72, 358 (1966)Google Scholar
  12. 12.
    Johnson, D.: A geometric form of Casson's invariant, and its connection to Reidemeister torsion. Unpublished lecture notesGoogle Scholar
  13. 13.
    Freed, D.: Reidemeister torsion, spectral sequences, and Breiskorn spheres. Preprint (University of Texas)Google Scholar
  14. 14.
    Fine, D.: Quantum Yang-Mills theory on the two sphere. Commun. Math. Phys.134, 273 (1990); Quantum Yang-Mills on a Riemann surface. Commun. Math. Phys.140, 321–338 (1991)Google Scholar
  15. 15.
    Migdal, A.: Zh. Eksp. Teor. Fiz.69, 810 (1975) (Sov. Phys. Jetp.42, 413)Google Scholar
  16. 16.
    Schwarz, A.: The partition function of degenerate quadratic functional and Ray-Singer invariants. Lett. Math. Phys.2, 247 (1978)Google Scholar
  17. 17.
    Witten, E.: Topology-changing amplitudes in 2+1 dimensional gravity. Nucl. Phys.323, 113 (1989)Google Scholar
  18. 18.
    Wilson, K.: Phys. Rev. D10, 2445 (1974)Google Scholar
  19. 19.
    Brocker, T., tom Dieck, T.: Representations of compact Lie groups. Berlin, Heidelberg, New York: Springer 1985Google Scholar
  20. 20.
    Atiyah, M.F.: Geometry and physics of knots. Cambridge: Cambridge University Press 1990Google Scholar
  21. 21.
    Jimbo, M., Miwa, T., Okada, M.: Lett. Math. Phys.14, 123 (1987); Mod. Phys. Lett. B1, 73 (1987); Commun. Math. Phys.116, 507 (1988)Google Scholar
  22. 22.
    Bralic, N.: Phys. Rec. D22, 3090 (1980)Google Scholar
  23. 23.
    Kazakov, V., Kostov, I.: Nucl. Phys. B176, 199 (1980); Phys. Lett. B105, 453 (1981); Nucl. Phys. B179, 283 (1981)Google Scholar
  24. 24.
    Gross, L., King, C., Sengupta, A.: Ann. Phys.194, 65 (1989)Google Scholar
  25. 25.
    Witten, E.: Gauge theories and integrable lattice models. Nucl. Phys. B322, 629 (1989)Google Scholar
  26. 26.
    Bott, R.: On E. Verlinde's formula in the context of stable bundles. Int. J. Mod. Phys.6, 2847 (1991)Google Scholar
  27. 27.
    Axelrod, S.: Ph. D. thesis (Princeton University, 1991)Google Scholar
  28. 27a.
    Axelrod, S., Witten, E.: UnpublishedGoogle Scholar
  29. 28.
    Cheeger, J.: Analytic torsion and the heat equation. Adv. Math.28, 233 (1978)Google Scholar
  30. 29.
    Muller, W.: Analytic torsion and theR-torsion of Riemannian manifolds. Ann. Math.109 (2), 259 (1979)Google Scholar
  31. 30.
    Bar-Natan, D., Witten, E.: Perturbative expansion of Chern-Simons theory with noncompact gauge group. Commun. Math. Phys. (to appear)Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Edward Witten
    • 1
  1. 1.School of Natural SciencesInstitute for Advanced StudyPrincetonUSA

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