On the D-Module and Formal-Variable Approaches to Vertex Algebras
In a program to formulate and develop two-dimensional conformal field theory in the framework of algebraic geometry, Beilinson and Drinfeld [BD] have recently given a notion of “chiral algebra” in terms of D-modules on algebraic curves. This definition consists of a “skew-symmetry” relation and a “Jacobi identity” relation in a categorical setting, and it leads to the operator product expansion for holomorphic quantum fields in the spirit of two-dimensional conformal field theory, as expressed in [BPZ], Because this operator product expansion, properly formulated, is known to be essentially a variant of the main axiom, the “Jacobi identity” [FLM], for vertex (operator) algebras ([Borc], [FLM]; see [FLM] for the proof), the chiral algebras of [BD] amount essentially to vertex algebras.
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- [BD]A. Beilinson and V. Drinfeld, unpublished manuscript.Google Scholar
- [Borel]A. Borel,Algebraic D-modules, Perspectives in Mathematics, Vol. 2, Academic Press, Boston, 1987.Google Scholar
- [FHL]I. B. Frenkel, Y.Z. Huang and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, preprint, 1989;Memoirs Amer. Math. Soc. 104, 1993.Google Scholar
- [FLM]I. B. Frenkel, J. Lepowsky and A. Meurman,Vertex Operator Alge¬bras and the Monster, Pure and Appl. Math., Vol. 134, Academic Press, Boston, 1988.Google Scholar
- [Ha]R. Hartshorne,Algebraic geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York, 1977.Google Scholar
- [Hul]YZ. Huang,On the geometric interpretation of vertex operator algebras, Ph.D. thesis, Rutgers University, 1990.Google Scholar
- [Hu2]YZ. Huang,Two-dimensional conformal geometry and vertex operator algebras, Birkhäuser, to appear.Google Scholar
- [HL1]YZ. Huang and J. Lepowsky, Operadic formulation of the notion of vertex operator algebra, in:Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups, Proc. Joint Summer Research Conference, Mount Holyoke, 1992, ed. P. Sally,M. Flato, J. Lepowsky, N. Reshetikhin and G. Zuckerman, Contemporary Math., Vol. 175, Amer. Math. Soc., Providence, 1994, 131 – 148.Google Scholar
- [HL2]YZ. Huang and J. Lepowsky, Tensor products of modules for a vertex operator algebras and vertex tensor categories, in:Lie Theory and Geometry, in honor of Bertram Kostant, ed. R. Brylinski, J.L. Brylinski, V. Guillemin, V. Kac, Birkhauser, Boston, 1994, 349 – 383.Google Scholar
- [HL3]Y–Z. Huang and J. Lepowsky, A theory of tensor products for module categories for a vertex operator algebra, I,Selecta Mathematica(New Series) 1 (1995).Google Scholar
- [HL4]Y–Z. Huang and J. Lepowsky, A theory of tensor products for module categories for a vertex operator algebra, II,Selecta Mathematica(New Series) 1 (1995).Google Scholar