The notion of vertex operator algebra arises naturally in the vertex operator construction of the Monster - the largest sporadic finite simple group. From another perspective, the theory of vertex operator algebras and their modules forms the algebraic foundation of conformal field theory. Vertex operator algebras and conformal field theory are now known to be deeply related to many important areas of mathematics. This essentially self-contained monograph develops the basic axiomatic theory of vertex operator algebras and their modules and intertwining operators, following a fundamental analogy with Lie algebra theory. The main axiom, the Jacobi(-Cauchy) identity, is a far-reaching analog of the Jacobi identity for Lie algebras. The authors show that the Jacobi identity is equivalent to suitably formulated rationality, commutativity, and associativity properties of products of quantum fields. A number of other foundational and useful results are also developed. This work was originally distributed as a preprint in 1989, and in view of the current widespread interest in the subject among mathematicians and theoretical physicists, its publication and availability should prove no less useful than when it was written.
|TAILLE DU FICHIER||1.99 MB|
|AUTEUR||Igor B. Frenkel|
|FICHIER||On Axiomatic Approaches to Vertex Operator Algebras and Modules.pdf|
The notion of vertex algebra due to Borcherds  and Frenkel-Lepowsky- Meurman ... Proof: The axioms we then need to verify are the version of equivariance for ... an operator L−1 : Vn → Vn+1, and an element 1 ∈ V0 such that the following ... fact  that VL is semisimple and its irreducible modules correspond to ... 10 Jul 2002 ... exposition of the algebro-geometric approach to vertex algebras and its ... review the mathematical development of vertex operator algebras and conformal ... axiom systems for vertex algebras, modules and intertwining ...