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Global vertex operators on Riemann surfaces

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Abstract

We develop an approach towards construction of conformal field theory starting from the basic axioms of vertex operator algebras.

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References

  • [A] Akman, F.: The semi-infinite Weil complex of a graded Lie algebra. Dissertation, Yale Univ., 1993

  • [Bo1] Borcherds, R.E.: Vertex operator algebras, Kac-Moody algebras and the Monster. Proc. Natl. Acad. Sci. USA.83, 3026 (1986)

    Google Scholar 

  • [Bo2] Borcherds, R.E.: Monstrous moonshine and monstrous Lie superalgebras. Invent. Math.109, 405 (1992)

    Article  Google Scholar 

  • [BPZ] Belavin, A., Polyakov, A.M., Zamolodchikv, A.A.: Infinite conformal symmetry in two dimensional quantum field theory. Nucl. Phys.B241, 33 (1984)

    Article  Google Scholar 

  • [BS] Belinson, A.A., Schechtman, V.V.: Determinant Bundles and Virasoro algebra. Commun. Math. Phys.118, 651–701 (1988)

    Article  Google Scholar 

  • [Co] Cohen, H.: Sums involving the values at negative integers ofL-functions of Quadratic characters. Math. Ann.217, 271–285 (1975)

    Article  Google Scholar 

  • [DVV] Dijkgraaf, R., Verlinde, E., Verlinde, H.:C=1 Conformal field theories on Riemann Surfaces. Commun. Math. Phys.115, 649–690 (1988)

    Article  Google Scholar 

  • [Do1] Dong, C.: Vertex algebras associated to even lattice. J. Alg., to appear (1990)

  • [Do2] Dong, C.: Representation of the moonshine module vertex operator algebra. Preprint (1992)

  • [Fa] Faltings, G.: A proof of the Verlinde Formula. Preprint (1992)

  • [FKW] Frenkel, E., Kac, V., Wakimoto, M.: Characters and Fusion Rules forW-algebras via Quantized Drinfeld-Sokolov Reductions. RIMS publication, Kyoto University861 (1992)

  • [FHL] Frenkel, I.B., Huang, Y., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Memoirs Am. Math., to appear

  • [FLM1] Frenkel, I.B., Lepowsky, J., Meurman, A.: A natural representation of the Fischer-Griess Monster with the modular functionJ as a character. Proc. Nat. Acad. Sci. USA81, 3256–3260 (1984)

    Google Scholar 

  • [FLM2] Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster. New York: Academic Press, 1988

    Google Scholar 

  • [FZ] Frenkel, I.B., Zhu, Y.: Vertex operator algebras associated to representation of affine and Virasoro algebras. Duke Math. J.66, 123 (1992)

    Article  Google Scholar 

  • [G] Gunning, R.C.: Lectures on Riemann surfaces. Princeton, NJ: Princeton Univ. Press, 1966

    Google Scholar 

  • [GGMV] Alvarez-Gaume, L., Gomez, C., Moore, G., Vafa, C.: Strings in the operator formalism. Nucl. Phys.B303, 455 (1988)

    Article  Google Scholar 

  • [H] Huang, Y.Z.: Geometric interpretation of vertex operator algebras. Proc. Natl. Acad. Sci. USA.88, 9964 (1991)

    Google Scholar 

  • [HL] Huang Y.Z., Lepowsky, J.: Toward a theory of tensor products for representations of a vertex operator algebra. In: Proc. 20th Intl. Conference on Differential geometric Methods in Theoretical Physics, Singapore: World Scientific, pp. 344–354

  • [KN] Krichever, I.M., Novikov, S.P.: Algebra of Virasoro type, Riemann surfaces and structure of the theory of solitons. Func. Anal. and its Appl.21, no. 2 (1987)

    Google Scholar 

  • [KNTY] Kawamoto, N., Namikawa, N., Tsuchiya, A., Yamada, Y.: Geometric realization of conformal field theory on Riemann surfaces. Commun. Math. Phys.116 (1988)

  • [Li] Lian, B.: On the classification of simple vertex operator algebras. Preprint (1992)

  • [Se] Segal, G.: Two dimensional conformal field theories and modular functors. In: IXth Proc. Internat. Cong. Math. Phys. Swansea 1988, Adam Hilger, 1989

  • [T] Tsuada, H.: String path integral realization of vertex operator algebras. Memoirs of Am. Math. Soc.91, no. 444 (1991)

    Google Scholar 

  • [TUY] Tsuchiya, A., Ueno, K., Yamada, Y.: Conformal field theory on universal family of stable curves with gauge symmetries. In: Advanced Studies on Pure Math., Vol.19, 1989, pp. 459–566.

  • [Wa] Wang, W.: Rationality of Virasoro vertex operator algebras. Preprint (1993)

  • [W] Witten, E.: Quantum field theory, Grassmannians, and algebraic curves. Commun. Math. Phys.113, 529–600 (1988)

    Article  Google Scholar 

  • [Z] Zhu, Y.: Vertex operators, elliptic functions and modular forms. Dissertation, Yale Univ., 1990

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Communicated by N.Yu. Reshetikhin

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Zhu, Y. Global vertex operators on Riemann surfaces. Commun.Math. Phys. 165, 485–531 (1994). https://doi.org/10.1007/BF02099421

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  • DOI: https://doi.org/10.1007/BF02099421

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