Introduction To Vertex Operator Algebras And Their Representations

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Introduction to Vertex Operator Algebras and Their Representations

James Lepowsky,Haisheng Li
Introduction to Vertex Operator Algebras and Their Representations Book PDF

Author : James Lepowsky,Haisheng Li
Publisher : Springer Science & Business Media
Page : 330 pages
File Size : 48,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9780817681869

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Introduction to Vertex Operator Algebras and Their Representations by James Lepowsky,Haisheng Li Pdf

* Introduces the fundamental theory of vertex operator algebras and its basic techniques and examples. * Begins with a detailed presentation of the theoretical foundations and proceeds to a range of applications. * Includes a number of new, original results and brings fresh perspective to important works of many other researchers in algebra, lie theory, representation theory, string theory, quantum field theory, and other areas of math and physics.

Spinor Construction of Vertex Operator Algebras, Triality, and E8(1)

Alex J. Feingold,Igor Frenkel,John F. X. Ries
Spinor Construction of Vertex Operator Algebras Triality and E8 1 Book PDF

Author : Alex J. Feingold,Igor Frenkel,John F. X. Ries
Publisher : American Mathematical Soc.
Page : 146 pages
File Size : 47,6 Mb
Release : 1991
Category : Mathematics
ISBN : 9780821851289

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Spinor Construction of Vertex Operator Algebras, Triality, and E8(1) by Alex J. Feingold,Igor Frenkel,John F. X. Ries Pdf

The theory of vertex operator algebras is a remarkably rich new mathematical field which captures the algebraic content of conformal field theory in physics. Ideas leading up to this theory appeared in physics as part of statistical mechanics and string theory. In mathematics, the axiomatic definitions crystallized in the work of Borcherds and in Vertex Operator Algebras and the Monster, by Frenkel, Lepowsky, and Meurman. The structure of monodromies of intertwining operators for modules of vertex operator algebras yields braid group representations and leads to natural generalizations of vertex operator algebras, such as superalgebras and para-algebras. Many examples of vertex operator algebras and their generalizations are related to constructions in classical representation theory and shed new light on the classical theory. This book accomplishes several goals. The authors provide an explicit spinor construction, using only Clifford algebras, of a vertex operator superalgebra structure on the direct sum of the basic and vector modules for the affine Kac-Moody algebra $D^{(1)}_n$. They also review and extend Chevalley's spinor construction of the 24-dimensional commutative nonassociative algebraic structure and triality on the direct sum of the three 8-dimensional $D_4$-modules. Vertex operator para-algebras, introduced and developed independently in this book and by Dong and Lepowsky, are related to one-dimensional representations of the braid group. The authors also provide a unified approach to the Chevalley, Griess, and $E_8$ algebras and explain some of their similarities. A third goal is to provide a purely spinor construction of the exceptional affine Lie algebra $E^{(1)}_8$, a natural continuation of previous work on spinor and oscillator constructions of the classical affine Lie algebras. These constructions should easily extend to include the rest of the exceptional affine Lie algebras. The final objective is to develop an inductive technique of construction which could be applied to the Monster vertex operator algebra. Directed at mathematicians and physicists, this book should be accessible to graduate students with some background in finite-dimensional Lie algebras and their representations. Although some experience with affine Kac-Moody algebras would be useful, a summary of the relevant parts of that theory is included. This book shows how the concepts and techniques of Lie theory can be generalized to yield the algebraic structures associated with conformal field theory. The careful reader will also gain a detailed knowledge of how the spinor construction of classical triality lifts to the affine algebras and plays an important role in a spinor construction of vertex operator algebras, modules, and intertwining operators with nontrivial monodromies.

Lie Algebras, Vertex Operator Algebras and Their Applications

Yi-Zhi Huang,Kailash C. Misra
Lie Algebras Vertex Operator Algebras and Their Applications Book PDF

Author : Yi-Zhi Huang,Kailash C. Misra
Publisher : American Mathematical Soc.
Page : 500 pages
File Size : 47,8 Mb
Release : 2007
Category : Lie algebras
ISBN : 9780821839867

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Lie Algebras, Vertex Operator Algebras and Their Applications by Yi-Zhi Huang,Kailash C. Misra Pdf

The articles in this book are based on talks given at the international conference 'Lie algebras, vertex operator algebras and their applications'. The focus of the papers is mainly on Lie algebras, quantum groups, vertex operator algebras and their applications to number theory, combinatorics and conformal field theory.

Generalized Vertex Algebras and Relative Vertex Operators

Chongying Dong,James Lepowsky
Generalized Vertex Algebras and Relative Vertex Operators Book PDF

Author : Chongying Dong,James Lepowsky
Publisher : Springer Science & Business Media
Page : 207 pages
File Size : 55,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461203537

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Generalized Vertex Algebras and Relative Vertex Operators by Chongying Dong,James Lepowsky Pdf

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

Igor Frenkel,James Lepowsky,Arne Meurman
Vertex Operator Algebras and the Monster Book PDF

Author : Igor Frenkel,James Lepowsky,Arne Meurman
Publisher : Academic Press
Page : 563 pages
File Size : 43,9 Mb
Release : 1989-05-01
Category : Mathematics
ISBN : 9780080874548

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Vertex Operator Algebras and the Monster by Igor Frenkel,James Lepowsky,Arne Meurman Pdf

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 Algebras and Algebraic Curves

Edward Frenkel,David Ben-Zvi
Vertex Algebras and Algebraic Curves Book PDF

Author : Edward Frenkel,David Ben-Zvi
Publisher : American Mathematical Soc.
Page : 418 pages
File Size : 43,7 Mb
Release : 2004-08-25
Category : Mathematics
ISBN : 9780821836743

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Vertex Algebras and Algebraic Curves by Edward Frenkel,David Ben-Zvi Pdf

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.

Vertex Operator Algebras and Related Areas

M. J. Bergvelt,Gaywalee Yamskulna,Wenhua Zhao
Vertex Operator Algebras and Related Areas Book PDF

Author : M. J. Bergvelt,Gaywalee Yamskulna,Wenhua Zhao
Publisher : American Mathematical Soc.
Page : 246 pages
File Size : 43,6 Mb
Release : 2009-10-01
Category : Mathematics
ISBN : 9780821848401

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Vertex Operator Algebras and Related Areas by M. J. Bergvelt,Gaywalee Yamskulna,Wenhua Zhao Pdf

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 Integral Bases for the Enveloping Algebras of Affine Lie Algebras

Shari A. Prevost
Vertex Algebras and Integral Bases for the Enveloping Algebras of Affine Lie Algebras Book PDF

Author : Shari A. Prevost
Publisher : American Mathematical Soc.
Page : 97 pages
File Size : 53,6 Mb
Release : 1992
Category : Mathematics
ISBN : 9780821825273

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Vertex Algebras and Integral Bases for the Enveloping Algebras of Affine Lie Algebras by Shari A. Prevost Pdf

We present a new proof of the identities needed to exhibit an explicit [bold]Z-basis for the universal enveloping algebra associated to an affine Lie algebra. We then use the explicit [bold]Z-bases to extend Borcherds' description, via vertex operator representations, of a [bold]Z-form of the enveloping algebras for the simply-laced affine Lie algebras to the enveloping algebras associated to the unequal root length affine Lie algebras.

Vertex Operators in Mathematics and Physics

J. Lepowsky,S. Mandelstam,I.M. Singer
Vertex Operators in Mathematics and Physics Book PDF

Author : J. Lepowsky,S. Mandelstam,I.M. Singer
Publisher : Springer Science & Business Media
Page : 484 pages
File Size : 41,7 Mb
Release : 2013-03-08
Category : Science
ISBN : 9781461395508

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Vertex Operators in Mathematics and Physics by J. Lepowsky,S. Mandelstam,I.M. Singer Pdf

James Lepowsky t The search for symmetry in nature has for a long time provided representation theory with perhaps its chief motivation. According to the standard approach of Lie theory, one looks for infinitesimal symmetry -- Lie algebras of operators or concrete realizations of abstract Lie algebras. A central theme in this volume is the construction of affine Lie algebras using formal differential operators called vertex operators, which originally appeared in the dual-string theory. Since the precise description of vertex operators, in both mathematical and physical settings, requires a fair amount of notation, we do not attempt it in this introduction. Instead we refer the reader to the papers of Mandelstam, Goddard-Olive, Lepowsky-Wilson and Frenkel-Lepowsky-Meurman. We have tried to maintain consistency of terminology and to some extent notation in the articles herein. To help the reader we shall review some of the terminology. We also thought it might be useful to supplement an earlier fairly detailed exposition of ours [37] with a brief historical account of vertex operators in mathematics and their connection with affine algebras. Since we were involved in the development of the subject, the reader should be advised that what follows reflects our own understanding. For another view, see [29].1 t Partially supported by the National Science Foundation through the Mathematical Sciences Research Institute and NSF Grant MCS 83-01664. 1 We would like to thank Igor Frenkel for his valuable comments on the first draft of this introduction.

Lie Algebras, Vertex Operator Algebras, and Related Topics

Katrina Barron,Elizabeth Jurisich,Antun Milas,Kailash Misr
Lie Algebras Vertex Operator Algebras and Related Topics Book PDF

Author : Katrina Barron,Elizabeth Jurisich,Antun Milas,Kailash Misr
Publisher : American Mathematical Soc.
Page : 274 pages
File Size : 42,8 Mb
Release : 2017-08-15
Category : Lie algebras
ISBN : 9781470426668

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Lie Algebras, Vertex Operator Algebras, and Related Topics by Katrina Barron,Elizabeth Jurisich,Antun Milas,Kailash Misr Pdf

This volume contains the proceedings of the conference on Lie Algebras, Vertex Operator Algebras, and Related Topics, celebrating the 70th birthday of James Lepowsky and Robert Wilson, held from August 14–18, 2015, at the University of Notre Dame, Notre Dame, Indiana. Since their seminal work in the 1970s, Lepowsky and Wilson, their collaborators, their students, and those inspired by their work, have developed an amazing body of work intertwining the fields of Lie algebras, vertex algebras, number theory, theoretical physics, quantum groups, the representation theory of finite simple groups, and more. The papers presented here include recent results and descriptions of ongoing research initiatives representing the broad influence and deep connections brought about by the work of Lepowsky and Wilson and include a contribution by Yi-Zhi Huang summarizing some major open problems in these areas, in particular as they pertain to two-dimensional conformal field theory.

Moonshine - The First Quarter Century and Beyond

James Lepowsky,John McKay,Michael P. Tuite
Moonshine The First Quarter Century and Beyond Book PDF

Author : James Lepowsky,John McKay,Michael P. Tuite
Publisher : Cambridge University Press
Page : 415 pages
File Size : 53,7 Mb
Release : 2010-06-03
Category : Mathematics
ISBN : 9780521106641

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Moonshine - The First Quarter Century and Beyond by James Lepowsky,John McKay,Michael P. Tuite Pdf

This volume examines the impact of the 'Monstrous Moonshine' paper on mathematics and theoretical physics.

Vertex Algebras for Beginners

Victor G. Kac
Vertex Algebras for Beginners Book PDF

Author : Victor G. Kac
Publisher : American Mathematical Soc.
Page : 209 pages
File Size : 46,5 Mb
Release : 1998
Category : Mathematical physics
ISBN : 9780821813966

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Vertex Algebras for Beginners by Victor G. Kac Pdf

Based on courses given by the author at MIT and at Rome University in spring 1997, this book presents an introduction to algebraic aspects of conformal field theory. It includes material on the foundations of a rapidly growing area of algebraic conformal theory.

Vertex Algebras and Geometry

Thomas Creutzig,Andrew R. Linshaw
Vertex Algebras and Geometry Book PDF

Author : Thomas Creutzig,Andrew R. Linshaw
Publisher : American Mathematical Soc.
Page : 168 pages
File Size : 52,7 Mb
Release : 2018-07-20
Category : Geometry, Algebraic
ISBN : 9781470437176

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Vertex Algebras and Geometry by Thomas Creutzig,Andrew R. Linshaw Pdf

This book contains the proceedings of the AMS Special Session on Vertex Algebras and Geometry, held from October 8–9, 2016, and the mini-conference on Vertex Algebras, held from October 10–11, 2016, in Denver, Colorado. The papers cover vertex algebras in connection with geometry and tensor categories, with topics in vertex rings, chiral algebroids, the Higgs branch conjecture, and applicability and use of vertex tensor categories.

From Vertex Operator Algebras to Conformal Nets and Back

Sebastiano Carpi,Yasuyuki Kawahigashi,Roberto Longo,Mihály Weiner
From Vertex Operator Algebras to Conformal Nets and Back Book PDF

Author : Sebastiano Carpi,Yasuyuki Kawahigashi,Roberto Longo,Mihály Weiner
Publisher : American Mathematical Soc.
Page : 85 pages
File Size : 54,9 Mb
Release : 2018-08-09
Category : Conformal invariants
ISBN : 9781470428587

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From Vertex Operator Algebras to Conformal Nets and Back by Sebastiano Carpi,Yasuyuki Kawahigashi,Roberto Longo,Mihály Weiner Pdf

The authors consider unitary simple vertex operator algebras whose vertex operators satisfy certain energy bounds and a strong form of locality and call them strongly local. They present a general procedure which associates to every strongly local vertex operator algebra V a conformal net AV acting on the Hilbert space completion of V and prove that the isomorphism class of AV does not depend on the choice of the scalar product on V. They show that the class of strongly local vertex operator algebras is closed under taking tensor products and unitary subalgebras and that, for every strongly local vertex operator algebra V, the map W↦AW gives a one-to-one correspondence between the unitary subalgebras W of V and the covariant subnets of AV.

Vertex Operator Algebras in Mathematics and Physics

Stephen Berman
Vertex Operator Algebras in Mathematics and Physics Book PDF

Author : Stephen Berman
Publisher : American Mathematical Soc.
Page : 265 pages
File Size : 51,9 Mb
Release : 2003
Category : Mathematical physics
ISBN : 9780821828564

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Vertex Operator Algebras in Mathematics and Physics by Stephen Berman Pdf

Vertex operator algebras are a class of algebras underlying a number of recent constructions, results, and themes in mathematics. These algebras can be understood as ''string-theoretic analogues'' of Lie algebras and of commutative associative algebras. They play fundamental roles in some of the most active research areas in mathematics and physics. Much recent progress in both physics and mathematics has benefited from cross-pollination between the physical and mathematical points of view. This book presents the proceedings from the workshop, ''Vertex Operator Algebras in Mathematics and Physics'', held at The Fields Institute. It consists of papers based on many of the talks given at the conference by leading experts in the algebraic, geometric, and physical aspects of vertex operator algebra theory. The book is suitable for graduate students and research mathematicians interested in the major themes and important developments on the frontier of research in vertex operator algebra theory and its applications in mathematics and physics.