On quantum gauge theories in two dimensions. (English) Zbl 0762.53063

Summary: Two dimensional quantum Yang-Mills theory is studied from three points of view: (i) by standard physical methods; (ii) by relating it to the large \(k\) 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.


53Z05 Applications of differential geometry to physics
81T13 Yang-Mills and other gauge theories in quantum field theory
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
Full Text: DOI


[1] Atiyah, M.F., Bott, R.: The Yang-Mills equations over Riemann surfaces. Phil. Trans. R. Soc. London A308, 523 (1982) · Zbl 0509.14014 · doi:10.1098/rsta.1983.0017
[2] Goldman, W.: The symplectic nature of fundamental groups of surfaces. Adv. Math.54, 200 (1984) · Zbl 0574.32032 · doi:10.1016/0001-8708(84)90040-9
[3] Ray, D., Singer, I.:R-torsion and the Laplacian on Riemannian manifolds. Adv. Math.7, 145 (1971) · Zbl 0239.58014 · doi:10.1016/0001-8708(71)90045-4
[4] Narasimhan, M.S., Seshadri, C.: Stable and unitary bundles on a compact Riemann surface. Ann. Math.82, 540 (1965) · Zbl 0171.04803 · doi:10.2307/1970710
[5] Verlinde, E.: Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys. B300, 360 (1988) · Zbl 1180.81120 · doi:10.1016/0550-3213(88)90603-7
[6] Tsuchiya, A., Ueno, K., Yamada, Y.: Conformal field theory on universal family of stable curves with gauge symmetries. Max-Planck-Institut preprint MPI/89/18 · Zbl 0696.17010
[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)
[8] Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.121, 351 (1989) · Zbl 0667.57005 · doi:10.1007/BF01217730
[9] Apostol, T.M.: Introduction to analytic number theory. Berlin, Heidelberg, New York: Springer 1984
[10] Thaddeus, M.: Conformal field theory and the moduli space of stable bundles. Oxford University preprint · Zbl 0772.53013
[11] Milnor, J.: Whitehead torsion. Bull. Am. Math. Soc.72, 358 (1966) · Zbl 0147.23104 · doi:10.1090/S0002-9904-1966-11484-2
[12] Johnson, D.: A geometric form of Casson’s invariant, and its connection to Reidemeister torsion. Unpublished lecture notes
[13] Freed, D.: Reidemeister torsion, spectral sequences, and Breiskorn spheres. Preprint (University of Texas) · Zbl 0743.57015
[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) · Zbl 0715.58046 · doi:10.1007/BF02097703
[15] Migdal, A.: Zh. Eksp. Teor. Fiz.69, 810 (1975) (Sov. Phys. Jetp.42, 413)
[16] Schwarz, A.: The partition function of degenerate quadratic functional and Ray-Singer invariants. Lett. Math. Phys.2, 247 (1978) · Zbl 0383.70017 · doi:10.1007/BF00406412
[17] Witten, E.: Topology-changing amplitudes in 2+1 dimensional gravity. Nucl. Phys.323, 113 (1989) · doi:10.1016/0550-3213(89)90591-9
[18] Wilson, K.: Phys. Rev. D10, 2445 (1974) · doi:10.1103/PhysRevD.10.2445
[19] Brocker, T., tom Dieck, T.: Representations of compact Lie groups. Berlin, Heidelberg, New York: Springer 1985
[20] Atiyah, M.F.: Geometry and physics of knots. Cambridge: Cambridge University Press 1990 · Zbl 0729.57002
[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) · Zbl 0642.17015 · doi:10.1007/BF00420302
[22] Bralic, N.: Phys. Rec. D22, 3090 (1980) · doi:10.1103/PhysRevD.22.3090
[23] Kazakov, V., Kostov, I.: Nucl. Phys. B176, 199 (1980); Phys. Lett. B105, 453 (1981); Nucl. Phys. B179, 283 (1981) · doi:10.1016/0550-3213(80)90072-3
[24] Gross, L., King, C., Sengupta, A.: Ann. Phys.194, 65 (1989) · Zbl 0698.60047 · doi:10.1016/0003-4916(89)90032-8
[25] Witten, E.: Gauge theories and integrable lattice models. Nucl. Phys. B322, 629 (1989) · doi:10.1016/0550-3213(89)90232-0
[26] Bott, R.: On E. Verlinde’s formula in the context of stable bundles. Int. J. Mod. Phys.6, 2847 (1991) · Zbl 0820.14024 · doi:10.1142/S0217751X91001404
[27] Axelrod, S.: Ph. D. thesis (Princeton University, 1991)
[28] Axelrod, S., Witten, E.: Unpublished
[29] Cheeger, J.: Analytic torsion and the heat equation. Adv. Math.28, 233 (1978) · Zbl 0395.57011 · doi:10.1016/0001-8708(78)90116-0
[30] Muller, W.: Analytic torsion and theR-torsion of Riemannian manifolds. Ann. Math.109 (2), 259 (1979) · Zbl 0412.58026 · doi:10.2307/1971113
[31] Bar-Natan, D., Witten, E.: Perturbative expansion of Chern-Simons theory with noncompact gauge group. Commun. Math. Phys. (to appear) · Zbl 0738.53041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.