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Symplectic structure of the moduli space of flat connection on a Riemann surface. (English) Zbl 0829.53028
The authors consider the canonical symplectic structure on the moduli space of flat 𝒢-connections on a Riemann surface of genus g with n marked points. A combinatorial description of this symplectic structure is given and its efficient formula is obtained for the case of a surface with marked points. The relation of the symplectic structure on and Poisson-Lie symplectic structures are studied. It is shown that for 𝒢 being a semisimple Lie algebra, the symplectic form may be represented as a sum of n copies of the Kirillov symplectic form on the orbit of dressing transformations in the Poisson- Lie group G * (corresponding to the Lie algebra 𝒢) and g copies of the symplectic structure on the Heisenberg double of the Poisson-Lie group G.

53C15Differential geometric structures on manifolds
58D27Moduli problems for differential geometric structures on spaces of mappings
[1]Alekseev, A.Yu., Malkin, A.Z.: Symplectic structures associated to Lie-Poisson groups. Commun. Math. Phys.162, 413 (1994) · Zbl 0797.58020 · doi:10.1007/BF02105190
[2]Arnold, V.I.: Mathematical methods of classical mechanics. Berlin, Heidelberg, New York: Springer, 1980
[3]Atiyah, M., Bott, R.: The Yang-Mills equations over a Riemann surface. Phil. Trans. R. Soc A308 523 (1982) · Zbl 0509.14014 · doi:10.1098/rsta.1983.0017
[4]Elitzur, S., Moore, G., Schwimmer, A., Seiberg, N.: Remarks on the canonical quantization of the Chern-Simons-Witten theory. Nucl. Phys.B 326, 108 (1989) · doi:10.1016/0550-3213(89)90436-7
[5]Fock, V.V., Rosly, A.A.: Poisson structure on the moduli space of flat connections on Riemann surfaces andr-matrix. Preprint ITEP 72-92, June 1992, Moscow
[6]Gawedzki, K., Falceto, F.: On quantum group symmetries of conformal field theories. Preprint IHES/P/91/59, September 1991
[7]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
[8]Kirillov, A.A.: Elements of the theory of representations. Berlin, Heidelberg, New York: Springer, 1976
[9]Lu, J.H., Weinstein, A.: Poisson-Lie groups, dressing transformations and Bruhat decompositions. J. Diff. Geom.,31, 501 (1990)
[10]Semenov-Tian-Shansky, M.A.: Dressing transformations and Poisson-Lie group actions. In: Publ. RIMS, Kyoto University21, no.6, 1237 (1985)
[11]Sklyanin, E.K.: Some algebraic structures related to the Yang-Baxter equation. Funk. Anal. i ego prilozh.,16, no. 4, 27–34 (1982)
[12]Witten, E.: Quantum Field Theory and the Jones Polynomial. Commun. Math. Phys.121, 351 (1989) · Zbl 0667.57005 · doi:10.1007/BF01217730
[13]Weinstein, A.: The local structure of Poisson manifolds. J. Diff. Geom.,18, n.3, 523–557 (1983)
[14]Weinstein, A.: The symplectic structure on moduli space. Preprint (1992)
[15]Guruprasad, K., Huebschmann, J., Jeffrey, L., Weinstein, A., to be published