Left-symmetric conformal algebras and vertex algebras

Abstract

A vertex algebra is an algebraic counterpart of a two-dimensional conformal field theory. By an equivalent characterization of vertex algebra using Lie conformal algebra and left-symmetric algebra given by Bakalov and Kac in [3], in studying vertex algebra, we have to deal with such a question: Do there exist compatible left-symmetric algebra structures on a class of special Lie algebras named formal distribution Lie algebras? In this paper, we study this question. We introduce the definitions of left-symmetric conformal algebra and Novikov conformal algebra. Many examples of these algebras are obtained. As an application, we present a construction of vertex algebra using left-symmetric conformal algebra.