Structure Theory of Finite Conformal Algebras

After the seminal paper [BPZ] of Belavin, Polyakov and Zamolodchikov, conformal field theory has become by now a large field with many remarkable ramifications to other fields of mathematics and physics. A rigorous mathematical definition of the “chiral part” of a conformal field theory, called a vertex (= chiral) algebra, was proposed by Borcherds [Bo] more than ten years ago and continued in [DL], [FHL], [FLM], [K], [L] and in numerous other works. However, until now a classification of vertex algebras, similar, for example, to the classification of finite-dimensional Lie algebras, seems to be far away. In the present paper we give a solution to the special case of this problem when the chiral algebra is generated by a finite number of quantum fields, closed under the operator product expansion (in the sense that only derivatives of the generating fields may occur). In this situation the adequate tool is the notion of a conformal algebra [K] which, to some extent, is related to a chiral algebra in the same way a Lie algebra is related to its universal enveloping algebra. At the same time, the theory of conformal algebras sheds a new light on the problem of classification of infinite-dimensional Lie algebras. About thirty years ago one of the authors posed (and partially solved) the problem of classification of simple Z-graded Lie algebras of finite Gelfand-Kirillov dimension [K1]. This problem was completely solved by Mathieu [M1]-[M3] in a remarkable tour de force. The point of view of the present paper is that the condition of locality (which is the most basic axiom of quantum field theory) along with a finiteness condition, are more natural conditions, which are also much easier to handle. In this paper we develop a structure theory of finite rank conformal algebras. Applications of this theory are two-fold. On the one hand, the conformal algebra structure is an axiomatic description [K] of the operator product expansion (OPE) of chiral fields in a conformal field theory [BPZ]. Hence the theory of finite conformal algebras provides a classification of finite systems of fields closed under the OPE. On the other hand, the category of finite conformal algebras is (more or less) equivalent to the category of infinite-dimensional Lie algebras spanned by Fourier coefficients of a finite number of pairwise local fields (or rather formal distributions)