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Geometric realization of conformal field theory on Riemann surfaces. (English) Zbl 0648.35080
This paper includes a self-contained account of the relation between the soliton theory (KP hierarchy), Grassmann manifolds and conformal field theory on Riemann surfaces.
The geometry of the (dressed) moduli space of Riemann surfaces and its embedding to the universal Grassmann manifold are described in algebro-geometrical way.
The theory of KP hierarchy, including the bosonization formalism, is completely reviewed. The vacuum state are given as the \(\tau\)-function for the compact solution (essentially Riemann’s theta function).
The correlation functions and the Ward identities are described. As the main result of this paper, it is proved that the Ward identities and some automorphy properties uniquely characterize the \(\tau\)-function.
Reviewer: N.Kawamoto

MSC:
35Q51 Soliton equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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[1] Alvarez-Gaumé, L., Moore, G., Vafa, C.: Theta functions, modular invariance, and strings. Commun. Math. Phys.106, 1-40 (1986) · Zbl 0605.58049
[2] Alvarez-Gaumé, L., Moore, G., Nelson, P., Vafa, C., Bost, J.B.: Bosonization in arbitrary genus. Phys. Lett.178, 41 (1986) · Zbl 0647.14019
[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) · Zbl 0647.14019
[4] Alvarez-Gaumé, L., Gomez, C., Reina, C.: Loop groups, grassmanians and string theory. Phys. Lett.190, 55 (1987)
[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)] · Zbl 0355.14002
[7] Belavin, A.A., Knizhnik, V.G.: Complex geometry and theory of quantum string. Sov. Phys. JETP64, 214 (1986) · Zbl 0693.58043
[8] Beilinson, A.A., Manin, Yu.I.: The Mumford form and the Polyakov measure in string theory. Commun. Math. Phys.107, 359 (1986) · Zbl 0604.14016
[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) · Zbl 0661.17013
[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)
[12] Bowick, M.J., Rajeev, S.G.: String theory as the Kähler geometry of loop space. Phys. Rev. Lett.58, 535 (1987)
[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 · Zbl 0571.35098
[14] Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Publ. Math. I.H.E.S.36, 75 (1969) · Zbl 0181.48803
[15] Dubrovin, B.A.: Theta functions and non-linear equations. Usp. Mat. Nauk36 (2), 11 (1981) · Zbl 0478.58038
[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)
[18] Eguchi, T., Ooguri, H.: Chiral bosonization on a Riemann surface. Phys. Lett.187, 127 (1987)
[19] Faltings, G.: Calculus on arithmetic surfaces. Ann. Math.119, 387 (1984) · Zbl 0559.14005
[20] Farkas, H.M., Kra, I.: Riemann surfaces. Berlin, Heidelberg, New York: Springer 1980 · Zbl 0475.30001
[21] Fay, J.: Theta functions on Riemann surfaces. Lecture Notes in Mathematics, Vol. 352. Berlin, Heidelberg, New York: Springer 1973 · Zbl 0281.30013
[22] Frenkel, I.B., Garland, H., Zuckerman, G.J.: Semi-infinite cohomology and string theory. Proc. Nat. Acad. Sci. USA83, 8442 (1986) · Zbl 0607.17007
[23] Friedan, D., Shenker, S.: The analytic geometry of two-dimensional conformal field theory. Nucl. Phys. B281, 509 (1987)
[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
[26] Hirota, R.: Direct method in soliton theory, Solitons, 157. Bullough, R.K., Caudrey, P.J. (eds.). Berlin, Heidelberg, New York: Springer 1980
[27] Ishibashi, N., Matsuo, Y., Ooguri, H.: Soliton equations and free fermions on Riemann surfaces. Mod. Phys. Lett. A2, 119 (1987)
[28] Ivanov, B.V.: On the quantum topology of strings. Phys. Lett.189, 39 (1987)
[29] Knizhnik, V.G.: Analytic fields on Riemann surfaces. Phys. Lett.180, 247 (1986) · Zbl 0656.58043
[30] Kodaira, K.: Complex manifolds and deformations of complex structure. Berlin, Heidelberg, New York: Springer 1985
[31] Krichever, I.M., Novikov, S.P.: Virasoro algebra, Riemann sirfaces and structure theory of soliton. Funct. Anal. Appl.21, 46 (1987) (in Russian) · Zbl 0634.17010
[32] Krichever, I.M.: Methods of algebraic geometry in the theory of non-linear equations. Russ. Math. Surv.32, 185 (1977) · Zbl 0386.35002
[33] Mulase, M.: Cohomological structure in soliton equations and jacobian varieties. J. Diff. Geom.19, 403 (1984) · Zbl 0559.35076
[34] Mumford, D.: Tata lectures on theta, Vols. I, II. Boston: Birkhäuser 1983 · Zbl 0509.14049
[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 · Zbl 0423.14007
[36] Mumford, D.: Curves and Jacobians. Ann Arbor, MI: The University of Michigan Press 1975 · Zbl 0316.14010
[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) · Zbl 0271.14014
[38] Peterson, D.H., Kac, V.: Infinite flag varieties and conjugacy theorems. Proc. Nat. Acad. Sci. USA,80, 1778 (1983) · Zbl 0512.17008
[39] Polyakov, A.M.: Quantum geometry of bosonic strings. Phys. Lett.103, 207 (1981)
[40] Sato, M.: Soliton equations as dynamical systems on an infinite dimensional Grassmann manifold. Res. Inst. Math. Sci., Kyoto Univ.-Kokyuroku439, 30 (1981) · Zbl 0507.58029
[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
[44] Segal, G.B., Wilson, G.: Loop groups and equations of KdV type, Publ. Math. I.H.E.S.61, 5 (1985) · Zbl 0592.35112
[45] Shiota, T.: Characterization of jacobian varieties in terms of soliton equations. Invent. Math. 333 (1986) · Zbl 0621.35097
[46] Siegel, C.L.: Topics of complex function theory, Vols. I, II. New York: Wiley · Zbl 0184.11201
[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)
[48] Tsuchiya, A., Kanie, Y.: Fock space representation of the Virasoro algebra-intertwining operators. Publ. R.I.M.S. Kyoto Univ.259, 22 (1986) · Zbl 0604.17008
[49] Tsuchiya, A., Kanie, Y.: Vertex operators on conformal field theory on ?1 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) · Zbl 0631.17010
[50] Vafa, C.: Operator formulation on Riemann surfaces. Phys. Lett.190, 47 (1987)
[51] Verlinde, E., Verlinde, H.: Chiral bosonization, determinants and string partition function. Nucl. Phys. B288, 357 (1987) · Zbl 0985.81681
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