Algebraic aspects of operator products and generalizations of vertex algebras

The notion of local quantum algebras that encode the "algebra" aspects of quantum field theories is introduced. This setup provides a foundation on which we can define precisely what an operator product expansion is, and we describe the relationships between local quantum algebras, operator product expansion, conformal field theories and some generalizations of vertex algebras that are introduced by various authors including Borcherds, Nikolov, and Huang and Kong.; Traditional approaches to vertex algebras are characterized by vertex operators and the formal calculus which lead to useful formal operator identities. We introduce the notion of factorization, and study local quantum algebras that satisfy factorization property in a general setting. Such a structure can be formulated abstractly, and we define the notion of graph algebras and study their relationships to vertex operators and derive some general identities.