Abstract
We present an analysis of the generalb−c system (including the β−γ system) on a compact Riemann surface of arbitrary genusg≧0 by postulating that its correlation functions should only have the singularities imposed by the operator product expansion (OPE) of the system. Studying a very (in fact optimally) general form of theb−c system, we prove rigorously that the standard practice of eliminating zero modes, and even the standard lagrangian, follow from the analyticity structure dictated by the OPE alone. We extend the analysis to consider the most general case of the presence of twist (e.g. spin) fields. We then determine all the possible correlation functions of theb−c system, with statistics unspecified, compatible with the OPE. On imposing Fermi and Bose statistics, we obtain the correlation functions of the fermionicb−c and β−γ systems, respectively.
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Friedan, D., Martinec, E., Shenker, S.: Conformal invariance, supersymmetry, and string theory. Nucl. Phys. B271, 93–165 (1986)
Eguchi, T., Ooguri, H.: Chiral bosonization on a Riemann surface. Phys. Lett. B187, 127–134 (1987)
Verlinde, E., Verlinde, H.: Chiral bosonization, determinants and the string partition function. Nucl. Phys. B288, 357–396 (1987)
Alvarez-Gaumé, L., Bost, J.-B., Moore, G., Nelson, P., Vafa, C.: Bosonization on higher genus Riemann surfaces. Commun. Math. Phys.112, 503–552 (1987)
Alvarez-Gaumé, L., Gomez, C., Reina, C.: New methods in string theory. In: Superstrings '87. Alvarez-Gaumé, L. (ed.). Singapore: World Scientific 1988
Kawamoto, N., Namikawa, Y., Tsuchiya, A., Yamada, Y.: Geometric realization of conformal field theory on Riemann surfaces. Commun. Math. Phys.116, 247–308 (1988)
Sen, S., Raina, A.K.: Grassmannians, multiplicative Ward identitites and theta function identities. Phys. Lett. B203, 256–262 (1988) Raina, A.K.: The divisor group and multiplicative Ward identities in conformal field theory. In: DST Workshop on particle physics, superstring theory. Ramachandran, R., Mani, H.S. (eds.). Singapore: World Scientific 1988
Atick, J.J., Sen, A.: Correlation functions of spin operators on a torus. Nucl. Phys. B286, 189–210 (1987)
Atick, J.J., Sen, A.: Spin field correlators on an arbitrary genus Riemann surface and nonrenormalization theorems in string theories. Phys. Lett. B186, 339–346 (1987)
Bonora, L., Lugo, A., Matone, M., Russo, J.: A global operator formalism on higher genus Riemann surfaces:b−c systems. Commun. Math. Phys.123, 329–352 (1989)
Semikhatov, A.M.: A global operator construction of theb−c system on Riemann surfaces in terms of a bosonic conformal theory. Phys. Lett. B212, 357–361 (1988)
Verlinde, E., Verlinde, H.: Multiloop calculations in covariant superstring theory. Phys. Lett. B192, 95–102 (1987) Semikhatov, A.M.: An operator formalism for the βγ systems on Riemann surfaces. Phys. Lett. B220, 406–412 (1989) Losev, A.: Once more on β, γ systems. Phys. Lett. B226, 62–66 (1989) Carow-Watamura, U., Ezawa, Z.F., Harada, K., Tezuka, A., Watamura, S.: Chiral bosonization of superconformal ghosts on the Riemann surface and path-integral measure. Phys. Lett. B227, 73–80 (1989) Lechtenfeld, O.: Superconformal ghost correlations on Riemann surfaces. Phys. Lett. B232, 193–198 (1989) Morozov, A.: A straightforward proof of Lechtenfeld's formula for the β, γ-correlator. Phys. Lett. B234, 15–17 (1990) Di Vecchia, P.: Correlation functions for the β−γ system from the sewing technique. Phys. Lett. B248, 329–334 (1990)
Raina, A.K.: Fay's trisecant identity and conformal field theory. Commun. Math. Phys.122, 625–641 (1989)
Raina, A.K.: Chiral fermions on a compact Riemann surface: a sheaf cohomology analysis. Lett. Math. Phys.19, 1–5 (1990)
Raina, A.K.: Fay's trisecant identity and Wick's theorem: an algebraic geometry viewpoint. Expo. Math.8, 227–245 (1990)
Raina, A.K.: Analyticity and chiral fermions on a Riemann surface. Helv. Phys. Acta63, 694–704 (1990)
Fay, J.D.: Theta functions on a Riemann surface. Berlin, Heidelberg, New York: Springer 1973
Cauchy, A.: Mémoire sur les fonctions alternées et sur les sommes alternées. Oeuvres Complètes, 2e série, t. 12, 173–182 [see Eq. (10) on p. 177] Paris: Gauthier-Villars et Cie 1916 Macdonald, I.G.: Symmetric functions and Hall polynomials. Oxford: Clarendon Press 1979
Frobenius, G.: Über die elliptischen Funktionen zweiter Art. J. Reine Angew. Math.93, 53–68 (1882) [see Eq. (12)]
Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley 1978
Hartshorne, R.: Algebraic geometry. Berlin, Heidelberg, New York: Springer 1977
Mumford, D.: Tata lectures on theta. II. Boston, Basel, Stuttgart: Birkhäuser 1984
Dixon, L., Friedan, D., Martinec, E., Shenker, S.: The conformal field theory of orbifolds. Nucl. Phys. B282, 13–73 (1987) Zucchini, R.: An operator formulation of orbifold conformal field theory. Commun. Math. Phys.129, 43–68 (1990)
Ramanujam, C.P.: Remarks on the Kodaira vanishing theorem. J. Indian Math. Soc.36, 41–51 (1972)
Littlewood, D.E., Richardson, A.R.: Group characters and algebra. Philos. Trans. R. Soc. London A233, 99–141 (1934) Littlewood, D.E.: The theory of group characters. Oxford: Clarendon Press 1940
Mumford, D.: Abelian varieties. Second edition. Bombay, London: TIFR and Oxford University Press 1974
Milne, J.S.: Abelian varieties. In: Arithmetic geometry. Cornell, G., Silverman, J.H. (eds.). Berlin, Heidelberg, New York: Springer 1986
Whittaker, E.T., Watson, G.N.: A course of modern analysis. Fourth edition reprinted. London, New York: Cambridge University Press 1969
Hejhal, D.: Theta functions, kernel functions and Abelian integrals. Mem. Am. Math. Soc.129 (1972)
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Raina, A.K. An algebraic geometry study of theb−c system with arbitrary twist fields and arbitrary statistics. Commun.Math. Phys. 140, 373–397 (1991). https://doi.org/10.1007/BF02099504
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DOI: https://doi.org/10.1007/BF02099504