Topics in Geometry pp 175-202 | Cite as

# On the D-Module and Formal-Variable Approaches to Vertex Algebras

## Abstract

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.

## Keywords

Open Subset Variable Approach Inverse Image Operator Product Expansion Jacobi Identity## Preview

Unable to display preview. Download preview PDF.

## References

- [BPZ]A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Infinite conformal symmetries in two-dimensional quantum field theory,
*Nucl. Phys.*B241 (1984), 333 – 380.MathSciNetCrossRefGoogle Scholar - [BD]A. Beilinson and V. Drinfeld, unpublished manuscript.Google Scholar
- [Borc]R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster,
*Proc. Natl. Acad. Sci. USA*83 (1986), 3068 – 3071.MathSciNetCrossRefGoogle 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