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Introduction to Vertex Operator Algebras and Their Representations

2004 Mathematics
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Release : 2004
Genre : Mathematics
Kind : eBook
Book Rating : 087/5 ( reviews)

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Book Synopsis Introduction to Vertex Operator Algebras and Their Representations by : James Lepowsky

Download or read book Introduction to Vertex Operator Algebras and Their Representations written by James Lepowsky. This book was released on 2004. Available in PDF, EPUB and Kindle. Book excerpt: The deep and relatively new field of vertex operator algebras is intimately related to a variety of areas in mathematics and physics: for example, the concepts of "monstrous moonshine," infinite-dimensional Lie theory, string theory, and conformal field theory. This book introduces the reader to the fundamental theory of vertex operator algebras and its basic techniques and examples. Beginning with a detailed presentation of the theoretical foundations and proceeding to a range of applications, the text includes a number of new, original results and also highlights and brings fresh perspective to important works of many researchers. After introducing the elementary "formal calculus'' underlying the subject, the book provides an axiomatic development of vertex operator algebras and their modules, expanding on the early contributions of R. Borcherds, I. Frenkel, J. Lepowsky, A. Meurman, Y.-Z. Huang, C. Dong, Y. Zhu and others. The concept of a "representation'' of a vertex (operator) algebra is treated in detail, following and extending the work of H. Li; this approach is used to construct important families of vertex (operator) algebras and their modules. Requiring only a familiarity with basic algebra, Introduction to Vertex Operator Algebras and Their Representations will be useful for graduate students and researchers in mathematics and physics. The booka??s presentation of the core topics will equip readers to embark on many active research directions related to vertex operator algebras, group theory, representation theory, and string theory.

Generalized Vertex Algebras and Relative Vertex Operators

2012-12-06 Mathematics
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Release : 2012-12-06
Genre : Mathematics
Kind : eBook
Book Rating : 538/5 ( reviews)

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Book Synopsis Generalized Vertex Algebras and Relative Vertex Operators by : Chongying Dong

Download or read book Generalized Vertex Algebras and Relative Vertex Operators written by Chongying Dong. This book was released on 2012-12-06. Available in PDF, EPUB and Kindle. Book excerpt: 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. 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.

Vertex Operator Algebras and the Monster

1989-04-11 Mathematics
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Release : 1989-04-11
Genre : Mathematics
Kind : eBook
Book Rating : 657/5 ( reviews)

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Book Synopsis Vertex Operator Algebras and the Monster by : Igor Frenkel

Download or read book Vertex Operator Algebras and the Monster written by Igor Frenkel. This book was released on 1989-04-11. Available in PDF, EPUB and Kindle. Book excerpt: This work is motivated by and develops connections between several branches of mathematics and physics--the theories of Lie algebras, finite groups and modular functions in mathematics, and string theory in physics. The first part of the book presents a new mathematical theory of vertex operator algebras, the algebraic counterpart of two-dimensional holomorphic conformal quantum field theory. The remaining part constructs the Monster finite simple group as the automorphism group of a very special vertex operator algebra, called the "moonshine module" because of its relevance to "monstrous moonshine."

Vertex Operator Algebras and Related Areas

2009-10-01 Mathematics
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Release : 2009-10-01
Genre : Mathematics
Kind : eBook
Book Rating : 402/5 ( reviews)

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Book Synopsis Vertex Operator Algebras and Related Areas by : M. J. Bergvelt

Download or read book Vertex Operator Algebras and Related Areas written by M. J. Bergvelt. This book was released on 2009-10-01. Available in PDF, EPUB and Kindle. Book excerpt: Vertex operator algebras were introduced to mathematics in the work of Richard Borcherds, Igor Frenkel, James Lepowsky and Arne Meurman as a mathematically rigorous formulation of chiral algebras of two-dimensional conformal field theory. The aim was to use vertex operator algebras to explain and prove the remarkable Monstrous Moonshine conjectures in group theory. The theory of vertex operator algebras has now grown into a major research area in mathematics. These proceedings contain expository lectures and research papers presented during the international conference on Vertex Operator Algebras and Related Areas, held at Illinois State University in Normal, IL, from July 7 to July 11, 2008. The main aspects of this conference were connections and interactions of vertex operator algebras with the following areas: conformal field theories, quantum field theories, Hopf algebra, infinite dimensional Lie algebras, and modular forms. This book will be useful for researchers as well as for graduate students in mathematics and physics. Its purpose is not only to give an up-to-date overview of the fields covered by the conference but also to stimulate new directions and discoveries by experts in the areas.

Vertex Algebras and Algebraic Curves

2004-08-25 Mathematics
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Release : 2004-08-25
Genre : Mathematics
Kind : eBook
Book Rating : 749/5 ( reviews)

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Book Synopsis Vertex Algebras and Algebraic Curves by : Edward Frenkel

Download or read book Vertex Algebras and Algebraic Curves written by Edward Frenkel. This book was released on 2004-08-25. Available in PDF, EPUB and Kindle. Book excerpt: Vertex algebras are algebraic objects that encapsulate the concept of operator product expansion from two-dimensional conformal field theory. Vertex algebras are fast becoming ubiquitous in many areas of modern mathematics, with applications to representation theory, algebraic geometry, the theory of finite groups, modular functions, topology, integrable systems, and combinatorics. This book is an introduction to the theory of vertex algebras with a particular emphasis on the relationship with the geometry of algebraic curves. The notion of a vertex algebra is introduced in a coordinate-independent way, so that vertex operators become well defined on arbitrary smooth algebraic curves, possibly equipped with additional data, such as a vector bundle. Vertex algebras then appear as the algebraic objects encoding the geometric structure of various moduli spaces associated with algebraic curves. Therefore they may be used to give a geometric interpretation of various questions of representation theory. The book contains many original results, introduces important new concepts, and brings new insights into the theory of vertex algebras. The authors have made a great effort to make the book self-contained and accessible to readers of all backgrounds. Reviewers of the first edition anticipated that it would have a long-lasting influence on this exciting field of mathematics and would be very useful for graduate students and researchers interested in the subject. This second edition, substantially improved and expanded, includes several new topics, in particular an introduction to the Beilinson-Drinfeld theory of factorization algebras and the geometric Langlands correspondence.

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