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Quantization of Chern-Simons gauge theory with complex gauge group. (English) Zbl 0717.53074
The author studies the canonical quantization of Chern-Simons gauge theory in \(2+1\) dimensions considering the case in which the gauge group is a complex Lie group. The Lagrangian is defined on a three-dimensional manifold \(M=\Sigma *{\mathbb{R}}\), with \({\mathbb{R}}'\) being the time axis and \(\Sigma\) an oriented closed surface. The special case in which \(\Sigma\) is a genus one surface is also analyzed. The quantization is described in close analogy with the compact gauge group case and the gravitational interpretation of the SL(2,\({\mathbb{C}})\) theory is also considered.
Reviewer: V.Silveira

53C80 Applications of global differential geometry to the sciences
81T13 Yang-Mills and other gauge theories in quantum field theory
81T70 Quantization in field theory; cohomological methods
Full Text: DOI
[1] Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.121, 351 (1989); Gauge theories and integrable lattice models, Nucl. Phys.B322, 351 (1989); Gauge theories, vertex models, and quantum groups. Nucl. Phys.B330, 285 (1990) · Zbl 0667.57005
[2] Achúcarro, A., Townsend: A Chern-Simons actions for three dimensional anti-De Sitter supergravity theories. Phys. Lett.180B, 89 (1986)
[3] Rocek, M., van Nieuwenhuizen, P.: Class. Quantum Grav.3, 43 (1986) · Zbl 0582.53059
[4] Witten, E.: 2+1 dimensional gravity as an exactly soluble system. Nucl. Phys.B311, 46 (1988) · Zbl 1258.83032
[5] Witten, E.: Topology-Changing amplitudes in 2+1 dimensional gravity. Nucl. Phys.B323, 113 (1989)
[6] Carlip, S.: Exact quantum scattering in 2+1 dimensional gravity. Nucl. Phys.B324, 106 (1989)
[7] Knizhnik, V. G., Polyakov, A. M., Zamolodchikov, A. B.: Fractal structure of 2d gravity. Mod. Phys. Lett.A3, 319 (1988)
[8] Verlinde, H.: Conformal field theory, 2-D quantum gravity, and quantization of Teichmüller space, Princeton preprint PUPT-89/1140
[9] Elitzur, S., Moore, G., Schwimmer, A., Seiberg, N.: Remarks on the canonical quantization of the Chern-Simons-Witten theory. IAS preprint HEP-89/20
[10] Bos, M., Nair, V.: Coherent state quantization of Chern-Simons theory. Columbia University preprint (May, 1989)
[11] Axelrod, S., Della Pietra, S., Witten, E.: Geometric quantization of Chern-Simons gauge theory, to appear in Jour. Diff. Geom. (May, 1991). · Zbl 0697.53061
[12] Hitchin, N.: Flat connections and geometric quantization. Commun. Math. Phys.131, 347–380 (1990) · Zbl 0718.53021
[13] Kostant, B.: Orbits, Symplectic Structures, and Representation Theory. Proc. of the U.S.-Japan Seminar in Differential Geometry, Kyoto, Japan, 1965. On Certain Unitary Representations Which Arise From a Quantization Theory. Lecture Notes in Math., Vol6, Battelle Seattle Rencontres, Berlin, Heidelberg, New York: Springer 1970, p. 237. Orbits And Quantization Theory. Proc. Int. Congress of Mathematicians, Nice, 1970, p. 395. Quantization and Unitary Representations. Lecture Notes in Math. Vol170, Berlin, Heidelberg, New York: Springer 1970, p. 87. Line Bundles and the Prequantized Schrodinger Equation. Coll. Group Theoretical Methods in Physics, Centre de Physique Theorique, (Marseille, 1972) p. 81. Symplectic Spinors. Symposia Mathematica (Rome), Vol XIV (1974), p. 139. On the Definition of Quantization. Geometric Symplectique et Physique Mathématique, Coll. CNRS, No. 237 (Paris, 1975), p. 187. Quantization and Representation Theory. Proc. Oxford Conference on Group Theory and Physics, (Oxford, 1977); Kostant, B., Auslander, L.: Polarization and unitary representations of solvable Lie groups, Invent. Math. 753 (1971)
[14] Souriau, J.-M.: Quantification geometrique. Commun. Math. Phys.1, 374 (1966), Structures Des Systems Dynamiques, Paris: Dunod 1970 · Zbl 1148.81307
[15] Sniatycki, J.: Geometric quantization and quantum mechanics, Berlin, Heidelberg, New York: Springer 1980 · Zbl 0429.58007
[16] Woodhouse, N.: Geometric quantization. Oxford: Oxford University Press 1980 · Zbl 0458.58003
[17] Carlip, S., de Alwis, S. P.: Wormholes in 2+1 dimensions. IAS preprint HEP-89/52 (1989)
[18] Geoffrey Mess,: Flat Lorentz spacetimes. Preprint (to appear) · Zbl 1206.83117
[19] Hitchin, N.: The self-duality equations on a Riemann surface. Proc. London Math. Soc.3, 55, 59 (1987) · Zbl 0634.53045
[20] Ray, D., Singer, I. M.: R-Torsion and the Laplacian on Riemannian Manifolds. Adv. Math.7, 145 (1971). Analytic Torsion of Complex Manifolds. Ann. Math.98, 154 (1973) · Zbl 0239.58014
[21] Quillen, D.: Determinants of Cauchy-Riemann operators over a Riemann surface. Funct. Anal. Appl.19, 31 (1985) · Zbl 0603.32016
[22] Bismut, J., Freed, D.: The analysis of elliptic families. Commun. Math. Phys.106, 159 (1986) · Zbl 0657.58037
[23] Witten, E.: The central charge in three dimensions. In Physics and Mathematics of Strings, Brink, L., Friedan, D., Polyakov, A. M. (eds.) Singapore: World Scientific 1990 · Zbl 0767.17023
[24] Atiyah, M. F.: On framings of 3-manifolds. Oxford University preprint (1989), to appear in Topology. · Zbl 0716.57011
[25] Knizhnik, V. G., Zamolodchikov, A. B.: Current algebra and Wess-Zumino model in two dimensions. Nucl. Phys.B247, 83 (1984) · Zbl 0661.17020
[26] Belavin, A., Polyakov, A., Zamolodchikov, A.: Infinite conformal symmetry in twodimensional quantum field theory. Nucl. Phys.B241, 333 (1984) · Zbl 0661.17013
[27] Verlinde, E.: Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys.B 300, 360 (1988) · Zbl 1180.81120
[28] Moore, G., Seiberg, N.: Polynomial equations for rational conformal field theories. Phys. Lett.B212, 360 (1988) Classical and quantum conformal field theory. Nucl. Phys. B.
[29] Dijkgraaf, R., Vafa, C., Verlinde, E., Verlinde, H.: Operator algebra of orbifold models. Commun. Math. Phys.123, 485 (1989) · Zbl 0674.46051
[30] Dijkgraaf, R., Witten, E.: Topological gauge theories and group cohomology. Commun. Math. Phys.129, 393 (1990) · Zbl 0703.58011
[31] Jakobsen, H. P., Kac, V. G.: A new class of unitarizable highest weight representations of infinite dimensional Lie algebras, preprint · Zbl 0581.17009
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