Vertex Operator Algebras and the Monster

Author: Igor Frenkel,James Lepowsky,Arne Meurman

Publisher: Academic Press

ISBN: 9780080874548

Category: Mathematics

Page: 508

View: 6274

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."
Posted in Mathematics

Lie Algebras, Lie Superalgebras, Vertex Algebras and Related Topics

Author: Kailash C. Misra,Daniel K. Nakano,Brian J. Parshall

Publisher: American Mathematical Soc.

ISBN: 1470418444

Category: Group theory and generalizations

Page: 355

View: 7028

This book contains the proceedings of the 2012–2014 Southeastern Lie Theory Workshop Series held at North Carolina State University in April 2012, at College of Charleston in December 2012, at Louisiana State University in May 2013, and at University of Georgia in May 2014. Some of the articles by experts in the field survey recent developments while others include new results in representations of Lie algebras, and quantum groups, vertex (operator) algebras and Lie superalgebras.
Posted in Group theory and generalizations

Moonshine, the Monster, and Related Topics

Joint Research Conference on Moonshine, the Monster, and Related Topics, June 18-23, 1994, Mount Holyoke College, South Hadley, Massachusetts

Author: Chongying Dong,Geoffrey Mason

Publisher: American Mathematical Soc.

ISBN: 0821803859

Category: Mathematics

Page: 368

View: 787

'One of the great legacies of the classification of the finite simple groups is the existence of the Monster $\ldots$. Work of Borcherds and Frenkel-Lepowsky-Meurman led to the notion of a vertex (operator) algebra, which was seen to be the same as the chiral algebras used by physicists in conformal field theory $\ldots$. The connections with physics have proven to be invaluable, and it seems likely that another branch of mathematics whose origins are eerily similar to those of moonshine - that is, elliptic cohomology - will turn out to be very relevant too' - from the Preface.This volume contains the proceedings of a Joint Summer Research Conference held at Mount Holyoke College in June 1994. As perhaps the first conference proceedings devoted exclusively to the subject known as 'Moonshine', this work contains something for many mathematicians and physicists. It features: results concerning the monster simple group and other simple groups; connections with elliptic cohomology; connections with 2-dimensional conformal field theory; the role of operads; and, connections with modular functions. ""Much of Moonshine, the Monster, and Related Topics"" features new results not available anywhere else.
Posted in Mathematics

Introduction to Vertex Operator Algebras and Their Representations

Author: James Lepowsky,Haisheng Li

Publisher: Springer Science & Business Media

ISBN: 9780817634087

Category: Mathematics

Page: 318

View: 2878

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.
Posted in Mathematics

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

Author: Alex J. Feingold,Igor Frenkel,John F. X. Ries

Publisher: American Mathematical Soc.

ISBN: 0821851284

Category: Mathematics

Page: 146

View: 3133

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.
Posted in Mathematics

Vertex Operator Algebras and Related Areas

An International Conference in Honor of Geoffrey Mason's 60th Birthday : July 7-11, 2008, Illinois State University, Normal, Illinois

Author: M. J. Bergvelt,Gaywalee Yamskulna,Wenhua Zhao

Publisher: American Mathematical Soc.

ISBN: 0821848402

Category: Mathematics

Page: 225

View: 3481

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.
Posted in Mathematics

Sūri kagaku kōkyūroku

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: N.A

View: 5659

Posted in Mathematics

OPE- Algebras

Author: Markus Rosellen

Publisher: N.A

ISBN: N.A

Category: Operator algebras

Page: 143

View: 3979

Posted in Operator algebras

Beweise und Widerlegungen

Die Logik mathematischer Entdeckungen

Author: Imre Lakatos

Publisher: Springer-Verlag

ISBN: 3663001962

Category: Mathematics

Page: 163

View: 9942

Posted in Mathematics

Kac-Moody and Virasoro Algebras

A Reprint Volume for Physicists

Author: Peter Goddard,David Olive

Publisher: World Scientific

ISBN: 9789971504205

Category: Science

Page: 586

View: 9617

This volume reviews the subject of Kac-Moody and Virasoro Algebras. It serves as a reference book for physicists with commentary notes and reprints.
Posted in Science

Algebra Colloquium

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Algebra

Page: N.A

View: 7077

Posted in Algebra

Pluto

Urasawa x Tezuka; nach der Geschichte "Astro Boy - der grösste Roboter auf Erden"

Author: Takashi Nagasaki,Naoki Urasawa,Osamu Tezuka

Publisher: N.A

ISBN: 9783551713087

Category:

Page: 230

View: 1450

Posted in

Manifolds and Modular Forms

Author: Friedrich Hirzebruch

Publisher: Springer-Verlag

ISBN: 3663140458

Category: Mathematics

Page: 212

View: 4416

Posted in Mathematics

On Axiomatic Approaches to Vertex Operator Algebras and Modules

Author: Igor Frenkel,Yi-Zhi Huang,James Lepowsky

Publisher: American Mathematical Soc.

ISBN: 0821825550

Category: Mathematics

Page: 64

View: 7924

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.
Posted in Mathematics

The Monster and Lie Algebras

Proceedings of a Special Research Quarter at the Ohio State University, May 1996

Author: Joseph Ferrar,Koichiro Harada

Publisher: Walter de Gruyter

ISBN: 3110801892

Category: Mathematics

Page: 262

View: 4946

This series is devoted to the publication of monographs, lecture resp. seminar notes, and other materials arising from programs of the OSU Mathemaical Research Institute. This includes proceedings of conferences or workshops held at the Institute, and other mathematical writings.
Posted in Mathematics