The Cocycle Condition of Liouville Theory and Its Application to the 2-D Induced Gravity(Ⅰ)

  • The cocycle condition of the action in the Liouville theory has been proposed.Several composition laws could be deduced from the cocycle condition,such as the composition law of the action in 2-D induced gravity in light-cone gauge,and that of the geometric action in coadjoint Deffs1 orbits proposed by Alekseev and Shatashivilli.
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  • [1] A. M. Polyakov, Mod. Phys. Lett., A2(1987), 893. V. G. Knizhnik, A. M. Polyakov and A. B. Zamolod-chikov, Mod. Phys Lett, A3(1988), 819.[2] A. Alekseev and S. Shat[1] A. M. Polyakov, Mod. Phys. Lett., A2(1987), 893. V. G. Knizhnik, A. M. Polyakov and A. B. Zamolod-chikov, Mod. Phys Lett, A3(1988), 819.[2] A. Alekseev and S. Shatashivili, Nucl. Phys., B323(1989), 719.[3] A. M. Polyakov, Int. J. Mod. Phys., A5(1990), 833[4] M. Bershadshy and H. Ooguri, Commutr. Math. Phys., 126(1989), 49.[5] See for example: S. Aoyama, J. Julve, Phys. Lett. B243(1990), 57.[6] See for example: F. David, Mod, Phys, Lett., A3(1988), 1651. J. Distler and H. Kawai,Nucl. Phys.,B321(1989), 509. N. Mavromatos and J. Miramontes, Mod, Phys.Lett, A4(1989), 1849. E. D'Hoker,P. S. Kurzepa 2-d quantum gravity and Liouville theory. UCLA/90/TEP/15[7] A. M. Polyakov, Phys. Lea., 103B(1981),207.[8] H. Verlinde, Preprint PUPT-89/1140, Sep.1989. H. Verlinde and E. Verlinde, Preprint PUPT-89/1149,Oct. 1989.[9] H. Y. Guo, J. M.Shen, S. K. Wang and K. W. Xu, Beltrami algebra and its operator formalism. Talk presented at the Beijing workshop on string theories, July 6-September 5, 1987, in proceedings ed. by X. C. Song; H. Y.Guo, J. M. Shen, S. K. Wang and K. W. Xu, Chinese Phys.Lett., 6(1989), 53; Beltra-mi algebra and symmetry of Beltrami eguatiort on Riemanrt surfaces to appear in J. Math. Phys.[10] S, K. Wang, Z. H. Wang, K. Wu and H. Y.Guo nonlinear conncctioc, Beltami connection and two dimensional gravity, to appear in Acta of Physics, H. Y. Guo,S. K. Wang, Z. H. Wang and K.wu,Commun. Theorc. Phys., 14(1990), 99.[11] D. Friedan, On recent advances in了field theory and statistical mechanics, Les Houches, 1982, ed by J. Zuber, R. Stora, (North-Holland) 839; H. Sonoda, Nucl. Phys., 8284(1987),157; E, D'Hoker and D. H. Phong, Rev. Mod. Phys., 60(1988), 917-1065.[12] S. Nag and A. Verjovsky, the coadyoint orbit spaces of Di介s1 and teichmuller space IC/89/290.[13] K. W. Xu and C. J. Zhu, Symntetry in two dimensional gravity, CTD-TAMU-56/90.[14] A. BeIavin, A, M. PoIyakov and A. Zamolodchikov, Nucl. Phys., 8241(1984), 333.[15] H. Verlinde and E. Verlinde, A Solution of Two Dimension Topological Quantum Gravity, Preprint PUPT-89/1176.[16] Z. H. Wang, K. Wu and H, Y. Guo, The cocycle condition of Liouville Theory and its application in2-d induccd gravity (Ⅱ).ashivili, Nucl. Phys., B323(1989), 719.[3] A. M. Polyakov, Int. J. Mod. Phys., A5(1990), 833[4] M. Bershadshy and H. Ooguri, Commutr. Math. Phys., 126(1989), 49.[5] See for example: S. Aoyama, J. Julve, Phys. Lett. B243(1990), 57.[6] See for example: F. David, Mod, Phys, Len., A3(1988), 1651. J. Distler and H. Kawai,Nucl. Phys.,B321(1989), 509. N. Mavromatos and J. Miramontes, Mod, Phys.Lett, A4(1989), 1849. E. D'Hoker,P. S. Kurzepa 2-d quantum gravity and Liouville theory. UCLA/90/TEP/15[7] A. M. Polyakov, Phys. Lea., 103B(1981),207.[8] H. Verlinde, Preprint PUPT-89/1140, Sep.1989. H. Verlinde and E. Verlinde, Preprint PUPT-89/1149,Oct. 1989.[9] H. Y. Guo, J. M.Shen, S. K. Wang and K. W. Xu, Beltrami algebra and its operator formalism. Talk presented at the Beijing workshop on string theories, July 6-September 5, 1987,in proceedings ed. by X. C. Song; H. Y.Guo, J. M. Shen, S. K. Wang and K. W. Xu, Chinese Phys.Lett., 6(1989), 53; Beltra-mi algebra and symmetry of Beltrami eguatiort on Riemanrt surfaces to appear in J. Math. Phys.[10] S, K. Wang, Z. H. Wang, K. Wu and H. Y.Guo nonlinear conncctioc, Beltami connection and two dimensional gravity, to appear in Acta of Physics, H. Y. Guo,S. K. Wang, Z. H. Wang and K.wu,Commun. Theorc. Phys., 14(1990), 99.[11] D. Friedan, On recent advances in了field theory and statistical mechanics, Les Houches, 1982, ed by J. Zuber, R. Stora, (North-Holland) 839; H. Sonoda, Nucl. Phys., B284(1987),157; E. D'Hoker and D. H. Phong, Rev. Mod. Phys., 60(1988), 917-1065.[12] S. Nag and A. Verjovsky, the coadyoint orbit spaces of Diffs1 and teichmuller space IC/89/290.[13] K. W. Xu and C. J. Zhu, Symntetry in two dimensional gravity, CTD-TAMU-56/90.[14] A. BeIavin, A, M. PoIyakov and A. Zamolodchikov, Nucl. Phys., B241(1984), 333.[15] H. Verlinde and E. Verlinde, A Solution of Two Dimension Topological Quantum Gravity, Preprint PUPT-89/1176[16] Z. H. Wang, K. Wu and H, Y. Guo, The cocycle condition of Liouville Theory and its application in2-d induccd gravity (Ⅱ).
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WANG Zhong-Hua, WU Ke and GUO Han-Ying. The Cocycle Condition of Liouville Theory and Its Application to the 2-D Induced Gravity(Ⅰ)[J]. Chinese Physics C, 1991, 15(9): 784-796.
WANG Zhong-Hua, WU Ke and GUO Han-Ying. The Cocycle Condition of Liouville Theory and Its Application to the 2-D Induced Gravity(Ⅰ)[J]. Chinese Physics C, 1991, 15(9): 784-796. shu
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The Cocycle Condition of Liouville Theory and Its Application to the 2-D Induced Gravity(Ⅰ)

    Corresponding author: WANG Zhong-Hua,
  • CCASTWorld Laboratory,Institute of Theoretical Physics,Academia Sinica,Beijing 100080

Abstract: The cocycle condition of the action in the Liouville theory has been proposed.Several composition laws could be deduced from the cocycle condition,such as the composition law of the action in 2-D induced gravity in light-cone gauge,and that of the geometric action in coadjoint Deffs1 orbits proposed by Alekseev and Shatashivilli.

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