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Geometric realization of conformal field theory on Riemann surfaces

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Abstract

Conformal field theory on a family of Riemann surfaces is formulated. We derive equations of motion of vacua which are parametrized by moduli of Riemann surfaces and show that these vacua are characterized uniquely by these equations. Our theory has a deep connection with Sato's theory of KP equations.

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References

  1. Alvarez-Gaumé, L., Moore, G., Vafa, C.: Theta functions, modular invariance, and strings. Commun. Math. Phys.106, 1–40 (1986)

    Google Scholar 

  2. Alvarez-Gaumé, L., Moore, G., Nelson, P., Vafa, C., Bost, J.B.: Bosonization in arbitrary genus. Phys. Lett.178, 41 (1986)

    Google Scholar 

  3. Alvarez-Gaumé, L., Bost, J.-B., Moore, G., Nelson, Ph., Vafa, C.: Bosonization on higher genus Riemann surfaces. Commun. Math. Phys.112, 503 (1987)

    Google Scholar 

  4. Alvarez-Gaumé, L., Gomez, C., Reina, C.: Loop groups, grassmanians and string theory. Phys. Lett.190, 55 (1987)

    Google Scholar 

  5. Alvarez, O., Windey, P.: The energy momentum tensor as geometrical datum, LBL preprint, UCB-PTH-86/36, LBL-22558

  6. Arakelov, S.: An intersection theory for divisors on an arithmetic surface. Izv. Akad. Nauk. SSSR Ser. Mat.38 (1974) [=Math. USSR Izv.8, 1167 (1974)]

    Google Scholar 

  7. Belavin, A.A., Knizhnik, V.G.: Complex geometry and theory of quantum string. Sov. Phys. JETP64, 214 (1986)

    Google Scholar 

  8. Beilinson, A.A., Manin, Yu.I.: The Mumford form and the Polyakov measure in string theory. Commun. Math. Phys.107, 359 (1986)

    Google Scholar 

  9. Beilinson, A.A., Manin, Yu.I., Shechtman, Y.A.: Localization of the Virasoro and Neveu-Schwartz algebra. Moscow preprint (1986)

  10. Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B241, 333 (1984)

    Google Scholar 

  11. Bernshtein, I.N., Rozenfel'd, B.I.: Homogeneous spaces of infinite-dimensional Lie algebra and characteristic classes of foliations. Russ. Math. Surv.107, 28 (1973)

    Google Scholar 

  12. Bowick, M.J., Rajeev, S.G.: String theory as the Kähler geometry of loop space. Phys. Rev. Lett.58, 535 (1987)

    Google Scholar 

  13. Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for soliton equations. In: Procs. of RIMS Symposium on Non-Linear Integrable Systems-Classical Theory and Quantum Theory, Kyoto, Japan. Jimbo, M., Miwa, T. (eds.). Singapore: World Science 1983

    Google Scholar 

  14. Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Publ. Math. I.H.E.S.36, 75 (1969)

    Google Scholar 

  15. Dubrovin, B.A.: Theta functions and non-linear equations. Usp. Mat. Nauk36 (2), 11 (1981)

    Google Scholar 

  16. Dugan, M., Sonoda, H.: Functional determinants on Riemann surfaces. LBL preprint, LBL-22776, 1986

  17. Eguchi, T., Ooguri, H.: Conformal and current algebras on a general Riemann surface. Nucl. Phys. B282, 308 (1987)

    Google Scholar 

  18. Eguchi, T., Ooguri, H.: Chiral bosonization on a Riemann surface. Phys. Lett.187, 127 (1987)

    Google Scholar 

  19. Faltings, G.: Calculus on arithmetic surfaces. Ann. Math.119, 387 (1984)

    Google Scholar 

  20. Farkas, H.M., Kra, I.: Riemann surfaces. Berlin, Heidelberg, New York: Springer 1980

    Google Scholar 

  21. Fay, J.: Theta functions on Riemann surfaces. Lecture Notes in Mathematics, Vol. 352. Berlin, Heidelberg, New York: Springer 1973

    Google Scholar 

  22. Frenkel, I.B., Garland, H., Zuckerman, G.J.: Semi-infinite cohomology and string theory. Proc. Nat. Acad. Sci. USA83, 8442 (1986)

    Google Scholar 

  23. Friedan, D., Shenker, S.: The analytic geometry of two-dimensional conformal field theory. Nucl. Phys. B281, 509 (1987)

    Google Scholar 

  24. Friedan, D.: A new formulation of string theory, Second Nobel Symposium on Elementary Particle Physics, presented at Marstrand, Sweden (1986) and Symposium on Geometry and Topology in Field Theory, Espoo, Finland (1986)

  25. van der Geer: Schottky's problem. (Proc.) Arbeitstagung Bonn. Lecture Notes in Mathematics, Vol. 1111. Berlin, Heidelberg, New York: Springer 1984

    Google Scholar 

  26. Hirota, R.: Direct method in soliton theory, Solitons, 157. Bullough, R.K., Caudrey, P.J. (eds.). Berlin, Heidelberg, New York: Springer 1980

    Google Scholar 

  27. Ishibashi, N., Matsuo, Y., Ooguri, H.: Soliton equations and free fermions on Riemann surfaces. Mod. Phys. Lett. A2, 119 (1987)

    Google Scholar 

  28. Ivanov, B.V.: On the quantum topology of strings. Phys. Lett.189, 39 (1987)

    Google Scholar 

  29. Knizhnik, V.G.: Analytic fields on Riemann surfaces. Phys. Lett.180, 247 (1986)

    Google Scholar 

  30. Kodaira, K.: Complex manifolds and deformations of complex structure. Berlin, Heidelberg, New York: Springer 1985

    Google Scholar 

  31. Krichever, I.M., Novikov, S.P.: Virasoro algebra, Riemann sirfaces and structure theory of soliton. Funct. Anal. Appl.21, 46 (1987) (in Russian)

    Google Scholar 

  32. Krichever, I.M.: Methods of algebraic geometry in the theory of non-linear equations. Russ. Math. Surv.32, 185 (1977)

    Google Scholar 

  33. Mulase, M.: Cohomological structure in soliton equations and jacobian varieties. J. Diff. Geom.19, 403 (1984)

    Google Scholar 

  34. Mumford, D.: Tata lectures on theta, Vols. I, II. Boston: Birkhäuser 1983

    Google Scholar 

  35. Mumford, D.: An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg-de-Vries equation and related non-linear equations, Intern. Symp. on Algebraic Geometry, Kyoto, 1977, 115

  36. Mumford, D.: Curves and Jacobians. Ann Arbor, MI: The University of Michigan Press 1975

    Google Scholar 

  37. Namikawa, Y.: On the canonical holomorphic map from the moduli space of stable curves to the Igusa monodial transform. Nagoya Math. J.52, 197 (1973)

    Google Scholar 

  38. Peterson, D.H., Kac, V.: Infinite flag varieties and conjugacy theorems. Proc. Nat. Acad. Sci. USA,80, 1778 (1983)

    Google Scholar 

  39. Polyakov, A.M.: Quantum geometry of bosonic strings. Phys. Lett.103, 207 (1981)

    Google Scholar 

  40. Sato, M.: Soliton equations as dynamical systems on an infinite dimensional Grassmann manifold. Res. Inst. Math. Sci., Kyoto Univ.-Kokyuroku439, 30 (1981)

    Google Scholar 

  41. Sato, M.: Soliton equations and the infinite dimensional Grassmann manifolds, Lectures delivered at Univ. of Tokyo, Nagoya Univ. (Notes by M. Mulase in Japanese 1981–1982) and Kyoto Univ. (Notes by T. Umeda in Japanese 1984–1985)

  42. Sato, M., Noumi, M.: Soliton equation and universal Grassmann manifold. Sophia University Kokyuroku in Math.18 (1984) (in Japanese)

  43. Sato, M., Sato, Y.: Solition equations as dynamical systems on infinite dimensional Grassmann manifold, Lecture Notes in Num. Appl. Anal.5, 259 (1982), Nonlinear PDE in Applied Science. U.S.-Japan Seminar, Tokyo, 1982

    Google Scholar 

  44. Segal, G.B., Wilson, G.: Loop groups and equations of KdV type, Publ. Math. I.H.E.S.61, 5 (1985)

    Google Scholar 

  45. Shiota, T.: Characterization of jacobian varieties in terms of soliton equations. Invent. Math. 333 (1986)

  46. Siegel, C.L.: Topics of complex function theory, Vols. I, II. New York: Wiley

  47. Sonoda, H.: Calculation of a propagator on a Riemann surface LBL preprint, LBL-21877, The energy-momentum tensor on a Riemann surface. Nucl. Phys. B281, 546 (1987)

    Google Scholar 

  48. Tsuchiya, A., Kanie, Y.: Fock space representation of the Virasoro algebra-intertwining operators. Publ. R.I.M.S. Kyoto Univ.259, 22 (1986)

    Google Scholar 

  49. Tsuchiya, A., Kanie, Y.: Vertex operators on conformal field theory on ℙ1 and monodromy representations of Braid groups, To appear in: Conformal field theory and solvable lattice model. Advanced Studies in Pure Mathematics, Kinokuniya. Lett. Math. Phys.13, 303 (1987)

    Google Scholar 

  50. Vafa, C.: Operator formulation on Riemann surfaces. Phys. Lett.190, 47 (1987)

    Google Scholar 

  51. Verlinde, E., Verlinde, H.: Chiral bosonization, determinants and string partition function. Nucl. Phys. B288, 357 (1987)

    Google Scholar 

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Author notes

  1. Yukihiko Namikawa

    Present address: Department of Mathematics, Nagoya University, 464, Nagoya, Japan

  2. Yasuhiko Yamada

    Present address: National Laboratory for High Energy Physics (KEK), Tsukuba, 305, Ibaraki, Japan

Authors and Affiliations

  1. Department of Physics, Kyoto University, 606, Kyoto, Japan

    Noboru Kawamoto

  2. Max-Planck-Institut für Mathematik, D-5300, Bonn, Federal Republic of Germany

    Yukihiko Namikawa

  3. Department of Mathematics, Nagoya University, 464, Nagoya, Japan

    Akihiro Tsuchiya

  4. Department of Physics, Nagoya University, 464, Nagoya, Japan

    Yasuhiko Yamada

Authors
  1. Noboru Kawamoto
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  2. Yukihiko Namikawa
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  3. Akihiro Tsuchiya
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  4. Yasuhiko Yamada
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Communicated by H. Araki

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Kawamoto, N., Namikawa, Y., Tsuchiya, A. et al. Geometric realization of conformal field theory on Riemann surfaces. Commun.Math. Phys. 116, 247–308 (1988). https://doi.org/10.1007/BF01225258

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  • DOI: https://doi.org/10.1007/BF01225258

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