Abstract
This paper is devoted in part to clarifying some aspects of the relation between quantum field theory and infinite Grassmannians, and in part to pointing out the existence of a close analogy between conformal field theory on Riemann surfaces and the modern theory of automorphic representations. Along the way we develop a multiplicative analog of the usual additive Ward identities of current algebra. We also reformulate the additive Ward identities in a way which may be useful, in terms of the residues of operator-valued differential forms. A concluding section is devoted to some remarks on string field theory. In an appendix, we attempt to clarify the recent construction by Beilinson, Manin, and Schechtman of what might be called global Virasoro algebras.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Segal, G., Wilson, G.: Loop groups and equations of KdV Type. Pub. IHES61 (1985)
Mulase, M.: J. Differ. Geom.19, 403 (1984)
Shiota, T.: Invent. Math.83, 333 (1986)
Ishibashi, N., Matsuo, Y., Ooguri, H.: Univ. of Tokyo preprint UT-499 (1986)
Alvarez-Gaumé, L., Gomez, C., Reina, C.: CERN TH-4641/87
Vafa, C.: Operator formulation on Riemann surfaces. Harvard preprint HUTP-87/A008
Gelbart, S.S.: Automorphic forms on Adele groups. Princeton, NJ: Princeton University Press 1975
An elementary introduction to the Langlands program. Bull. AMS10, 177 (1984)
Ray, D., Singer, I.: Analytic torsion. Ann. Math. (1973)
Quillen, D.: Determinants of Cauchy-Riemann operators over a Riemann surface. Funkts. Anal. Prilozh.19, 37 (1985) (Funct. Anal. Appl.19, 31 (1986))
Atiyah, M.F., Singer, I.M.: Proc. Natl. Acad. Sci.81, 2597 (1984)
Atiyah, M.F., Patodi, V.K., Singer, I.: Proc. Camb. Philos. Soc.77, 43 (1975);78, 405 (1975);79, 71 (1976)
Pressley, A., Segal, G.B.: Loop groups and their representations. Oxford: Oxford University Press 1986
Bost, J., Nelson, P.: Spin 1/2 Bosonization on compact surfaces. Phys. Rev. Lett.57, 795 (1986)
Alvarez Gaumé, L., Bost, J.B., Moore, G., Nelson, P., Vafa, C.: Bosonization in arbitrary genus. Phys. Lett.B 178, 41 (1986)
Bosonization on higher genus Riemann surfaces. Commun. Math. Phys. (to appear)
Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys.B 241, 333 (1984)
Friedan, D., Martinec, E., Shenker, S.: Conformal invariance, supersymmetry and string theory. Nucl. Phys.B 271, 93 (1986)
Frenkel, I.: Spinor representations of affine Lie algebras. Proc. Natl. Acad. Sci. USA77, 6303 (1980)
Two constructions of affine Lie algebra representations. J. Funct. Anal.44, 259 (1981)
Garland, H.: The arithmetic theory of loop algebras. J. Algebra53, 480 (1970)
Bowick, M.J., Rajeev, S.G.: MIT preprints (1987)
Beilinson, A.A., Manin, Ya.I., Shechtman, V.A.: Localization of the Virasoro and Neveu-Schwarz algebras. Preprint (1986)
Witten, E.: Noncommutative geometry and string field theory. Nucl. Phys.B 268, 253 (1986)
D'Hoker, E., Giddings, S.: Unitarity of the closed bosonic polyakov string. Princeton preprint PUPT-1046 (1987)
Strominger, A.: IAS preprints (1986 and 1987)
Author information
Authors and Affiliations
Additional information
Communicated by A. Jaffe
Supported in part by NSF Grants PHY80-19754 and 86-16129
Rights and permissions
About this article
Cite this article
Witten, E. Quantum field theory, Grassmannians, and algebraic curves. Commun.Math. Phys. 113, 529–600 (1988). https://doi.org/10.1007/BF01223238
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01223238