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Axioms for a vertex algebra and the locality of quantum fields. (English) Zbl 0928.17025
MSJ Memoirs. 4. Tokyo: Mathematical Society of Japan. ix, 110 p. (1999).
In this booklet basics in vertex algebras and two-dimensional chiral quantum fields are presented in a very formal manner. Starting in the first part with formal Laurent series over vector fields having infinitely many terms on both sides, fields and mutually local fields are defined by formal Laurent series satisfying special properties. Discussing algebraic operations with such Laurent series several identities are derived. But the reader is left alone with the questions whether and how these considerations are relateed to objects and procedures in physics having the same name. He has to know the physical background or to take it as a game in a mathematical concise but very dry exposition.
The main tool of the first part seems to be the identity which the authors call Borcherds identity. This part is organized as follows: I. Two-dimensional chiral quantum fields, 1. Fields and their residue products, 2. Mutually local fields, 3. Borcherds identity for local fields.
In the second part a definition of vertex algebra is given using Borcherds identity as a central part. Consequences of this definition and its relation to other definitions are discussed. The points are: II. Axioms for a vertex algebra, 4. Axioms and their consequences, 5. State field correspondence, 6. Goddard’s axioms and the existence theorem.
A third part contains in a rather short description some connections with other structures. In 8.1 f.i. it is pointed out on three pages how a vertex algebra gives rise to a Lie algebra structure on a suitable quotient space, but no deeper discussion takes place. There are various sketchy examples of vertex algebras. This part is organized by: III. Topics and examples, 7. Summary of related notions, 8. Relation to other algebraic structures, 9. Examples.
In an Appendix vertex superalgebras and admissible fields are defined: IV. Appendix, A. Vertex superalgebras, B. Analytic method, C. List of expansions of $$(x-y)^r(y-z)^q(x-z)^p$$.

##### MSC:
 17B69 Vertex operators; vertex operator algebras and related structures 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 17-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to nonassociative rings and algebras 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations
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