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. In this paper, we show directly that the chiral algebras of [BD] are essentially the same as vertex algebras without vacuum vector (and without grading), by establishing an equivalence between the skew-symmetry and Jacobi identity relations of [BD] and the (similarly-named, but different) skewsymmetry and Jacobi identity relations in the formal-variable approach to vertex operator algebra theory (see [FLM], [FHL]). In particular, among the equivalent formulations of the notion of vertex (operator) algebra, the D-module notion of chiral algebra corresponds the most closely to the formalvariable notion, rather than to, say, the operator-product-expansion notion (based on the “commutativity” and “associativity” relations, as explained in [FLM], [FHL]) or to the geometric or operadic notion ([Hu1], [Hu2], [HL1]). More precisely, we prove that for any nonempty open subset X of C, the category of vertex algebras without vacuum over X (see Definitions 2.2