Formal Distribution Algebras and Conformal Algebras

Conformal algebra is an axiomatic description of the operator product expansion (or rather its Fourier transform) of chiral fields in a conformal field theory. It turned out to be an adequate tool for the realization of the program of the study of Lie (super)algebras and associative algebras (and their representations), satisfying the sole locality property [K3]. The first basic definitions and results appeared in my book [K] and review [K3]. In the present paper I review recent developments in conformal algebra, including some of [K] and [K3] but in a different language. Here I use the λ-product, which is the Fourier transform of the OPE, or, equivalently, the generating series of the n-th products used in [K] and [K3]. This makes the exposition much more elegant and transparent. Most of the work has been done jointly with my collaborators. The structure theory of finite Lie conformal algebras is a joint paper with A. D’Andrea [DK]. The theory of conformal modules has been developed with S.-J. Cheng [CK] and of their extensions with S.-J. Cheng and M. Wakimoto [CKW]. The understanding of conformal algebras CendN and gcN was achieved with A. D’Andrea [DK], and of their finite representations with B. Bakalov, A. Radul and M. Wakimoto [BKRW]. The connection to Γ-local and Γ-twisted formal distribution algebras has been established with M. GolenishchevaKutuzova [GK] and with B. Bakalov and A. D’Andrea [BDK]. Cohomology theory has been worked out with B. Bakalov and A. Voronov [BKV].