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 special but 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.
