What is a vertex algebra?

In this paper we try to define the higher dimensional analogues of vertex algebras. In other words we define algebras which we hope have the same relation to higher dimensional quantum field theories that vertex algebras have to one dimensional quantum field theories (or to ``chiral halves'' of two dimensional quantum field theories). The main idea is to define "vertex groups". Then classical vertex algebras turn out to be the same as "associative commutative algebras" over t...

These lecture notes are intended to give a modest impulse to anyone willing to start or pursue a journey into the theory of Vertex Algebras by reading one of Kac's or Lepowsky-Li's books. Therefore, the primary goal is to provide required tools and help being acquainted with the machinery which the whole theory is based on. The exposition follows Kac's approach. Fundamental examples relevant in Theoretical Physics are also discussed. No particular prerequisites are assumed.

We characterize vertex algebras (in a suitable sense) as algebras over a certain graded co-operad. We also discuss some examples and categorical implications of this characterization.

This book offers an introduction to vertex algebra based on a new approach. The new approach says that a vertex algebra is an associative algebra such that the underlying Lie algebra is a vertex Lie algebra. In particular, vertex algebras can be formulated in terms of a single multiplication and they behave like associative algebras with respect to it. Chapter 1 is the introduction. In chapter 2 we discuss many examples of vertex Lie algebras and we show that vertex Lie alg...

In this exposition, I discuss several developments in the theory of vertex operator algebras, and I include motivation for the definition.

A higher dimensional analogue of the notion of vertex algebra is formulated in terms of formal variable language with Borcherds' notion of $G$-vertex algebra as a motivation. Some examples are given and certain analogous duality properties are proved. Furthermore, it is proved that for any vector space $W$, any set of mutually local multi-variable vertex operators on $W$ in a certain canonical way generates a vertex algebra with $W$ as a natural module.

This is the text of the Bourbaki seminar that I gave on June 24, 2000.

Vertex algebras provide an axiomatic algebraic description of the operator product expansion (OPE) of chiral fields in 2-dimensional conformal field theory. Vertex Lie algebras (= Lie conformal algebras) encode the singular part of the OPE, or, equivalently, the commutators of chiral fields. We discuss generalizations of vertex algebras and vertex Lie algebras, which are relevant for higher-dimensional quantum field theory.

We propose an extension of the definition of vertex algebras in arbitrary space-time dimensions together with their basic structure theory. An one-to-one correspondence between these vertex algebras and axiomatic quantum field theory (QFT) with global conformal invariance (GCI) is constructed.

Foundations of the theory of vertex algebras are extended to the non-Archimedean setting.