Generalized Vertex Algebras and Relative Vertex Operators

In the past few years, vertex operator algebra theory has been growing both in intrinsic interest and in the scope of its interconnections with areas of mathematics and physics. The structure and representation theory of vertex operator algebras is deeply related to such subjects as monstrous moonshine, conformal field theory and braid group theory. Vertex operator algebras are the mathematical counterpart of chiral algebras in conformal field theory. In the Introduction which follows, we sketch some of the main themes in the historical development and mathematical and physical motivations of these ideas, and some of the current issues. Given a vertex operator algebra, it is important to consider not only its modules (representations) but also intertwining operators among the mod­ ules. Matrix coefficients of compositions of these operators, corresponding to certain kinds of correlation functions in conformal field theory, lead natu­ rally to braid group representations. In the specialbut important case when these braid group representations are one-dimensional, one can combine the modules and intertwining operators with the algebra to form a structure satisfying axioms fairly close to those for a vertex operator algebra. These are the structures which form the main theme of this monograph. Another treatment of similar structures has been given by Feingold, Frenkel and Ries (see the reference [FFR] in the Bibliography), and in fact the material de­ veloped in the present work has close connections with much work of other people, as we explain in the Introduction and throughout the text.

The rapidly-evolving theory of vertex operator algebras provides deep insight into many important algebraic structures. Vertex operator algebras can be viewed as "complex analogues" of both Lie algebras and associative algebras. They are mathematically precise counterparts of what are known in physics as chiral algebras, and in particular, they are intimately related to string theory and conformal field theory.

Dong and Lepowsky have generalized the theory of vertex operator algebras in a systematic way at three successively more general levels, all of which incorporate one-dimensional braid groups representations intrinsically into the algebraic structure: First, the notion of "generalized vertex operator algebra" incorporates such structures as Z-algebras, parafermion algebras, and vertex operator superalgebras. Next, what they term "generalized vertex algebras" further encompass the algebras of vertex operators associated with rational lattices. Finally, the most generalof the three notions, that of "abelian intertwining algebra," also illuminates the theory of intertwining operator for certain classes of vertex operator algebras.

The monograph is written in a n accessible and self-contained manner, with detailed proofs and with many examples interwoven through the axiomatic treatment as motivation and applications. It will be useful for research mathematicians and theoretical physicists working the such fields as representation theory and algebraic structure sand will provide the basis for a number of graduate courses and seminars on these and related topics.