From Vertex Operator Algebras To Conformal Nets And Back

Author : Sebastiano Carpi,Yasuyuki Kawahigashi,Roberto Longo,Mihály Weiner
Publisher : American Mathematical Soc.
Page : 85 pages
File Size : 41,9 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.

Author : Yi-Zhi Huang,Kailash C. Misra
Publisher : American Mathematical Soc.
Page : 500 pages
File Size : 45,7 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.

Author : James Lepowsky,Haisheng Li
Publisher : Springer Science & Business Media
Page : 330 pages
File Size : 41,9 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.

Author : M. J. Bergvelt,Gaywalee Yamskulna,Wenhua Zhao
Publisher : American Mathematical Soc.
Page : 246 pages
File Size : 50,6 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.

Author : Katrina Barron,Elizabeth Jurisich,Antun Milas,Kailash Misr
Publisher : American Mathematical Soc.
Page : 274 pages
File Size : 55,5 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.

Author : Stephen Berman
Publisher : American Mathematical Soc.
Page : 268 pages
File Size : 41,5 Mb
Release : 2024-06-28
Category : Mathematics
ISBN : 0821871447

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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.

Author : Alex J. Feingold,Igor Frenkel,John F. X. Ries
Publisher : American Mathematical Soc.
Page : 146 pages
File Size : 55,7 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.

Author : Matthew Krauel,Michael Tuite,Gaywalee Yamskulna
Publisher : American Mathematical Soc.
Page : 250 pages
File Size : 46,5 Mb
Release : 2020-07-13
Category : Education
ISBN : 9781470449384

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Vertex Operator Algebras, Number Theory and Related Topics by Matthew Krauel,Michael Tuite,Gaywalee Yamskulna Pdf

This volume contains the proceedings of the International Conference on Vertex Operator Algebras, Number Theory, and Related Topics, held from June 11–15, 2018, at California State University, Sacramento, California. The mathematics of vertex operator algebras, vector-valued modular forms and finite group theory continues to provide a rich and vibrant landscape in mathematics and physics. The resurgence of moonshine related to the Mathieu group and other groups, the increasing role of algebraic geometry and the development of irrational vertex operator algebras are just a few of the exciting and active areas at present. The proceedings center around active research on vertex operator algebras and vector-valued modular forms and offer original contributions to the areas of vertex algebras and number theory, surveys on some of the most important topics relevant to these fields, introductions to new fields related to these and open problems from some of the leaders in these areas.

Author : Igor Frenkel
Publisher : Oxford University Press, USA
Page : 79 pages
File Size : 50,6 Mb
Release : 2014-08-31
Category : MATHEMATICS
ISBN : 1470400715

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On Axiomatic Approaches to Vertex Operator Algebras and Modules by Igor Frenkel Pdf

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.

Author : Chongying Dong,James Lepowsky
Publisher : Springer Science & Business Media
Page : 207 pages
File Size : 55,6 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.

Author : Thomas Creutzig,Shashank Kanade,Robert McRae
Publisher : American Mathematical Society
Page : 194 pages
File Size : 46,9 Mb
Release : 2024-04-17
Category : Mathematics
ISBN : 9781470467241

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Tensor Categories for Vertex Operator Superalgebra Extensions by Thomas Creutzig,Shashank Kanade,Robert McRae Pdf

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Author : Yi-Zhi Huang
Publisher : Unknown
Page : 300 pages
File Size : 47,7 Mb
Release : 1997-07-15
Category : Electronic
ISBN : 1461242770

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Two-Dimensional Conformal Geometry and Vertex Operator Algebras by Yi-Zhi Huang Pdf

Author : Romeo Brunetti,Claudio Dappiaggi,Klaus Fredenhagen,Jakob Yngvason
Publisher : Springer
Page : 453 pages
File Size : 47,9 Mb
Release : 2015-09-04
Category : Science
ISBN : 9783319213538

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Advances in Algebraic Quantum Field Theory by Romeo Brunetti,Claudio Dappiaggi,Klaus Fredenhagen,Jakob Yngvason Pdf

This text focuses on the algebraic formulation of quantum field theory, from the introductory aspects to the applications to concrete problems of physical interest. The book is divided in thematic chapters covering both introductory and more advanced topics. These include the algebraic, perturbative approach to interacting quantum field theories, algebraic quantum field theory on curved spacetimes (from its structural aspects to the applications in cosmology and to the role of quantum spacetimes), algebraic conformal field theory, the Kitaev's quantum double model from the point of view of local quantum physics and constructive aspects in relation to integrable models and deformation techniques. The book is addressed to master and graduate students both in mathematics and in physics, who are interested in learning the structural aspects and the applications of algebraic quantum field theory.

Author : Patrick Dorey,Gregory Korschemsky,Nikita Nekrasov,Didina Serban
Publisher : Lecture Notes of the Les Houch
Page : 573 pages
File Size : 43,9 Mb
Release : 2019
Category : Mathematics
ISBN : 9780198828150

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Integrability by Patrick Dorey,Gregory Korschemsky,Nikita Nekrasov,Didina Serban Pdf

This volume, 106 of the Les Houches Summer School series, brings together applications of integrability to supersymmetric gauge and string theory. The book focuses on the application of integrability and problems in quantum field theory. Particular emphasis is given to the exact solution of planar N=4 super-Yang-Mills theory and its relation with string theory on the one hand, and the exact determination of the low-energy physics of N=2 super-Yang-Mills theories on the other; links with other domains are also explored. The purpose of the Les Houches Summer School was to bring together young researchers and specialists from statistical physics, condensed matter physics, gauge and string theory, and mathematics, to stimulate discussion across these different research areas.

Author : Victor G. Kac
Publisher : Unknown
Page : 160 pages
File Size : 49,9 Mb
Release : 1997
Category : Mathematics
ISBN : UOM:39015038539923

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Vertex Algebras for Beginners by Victor G. Kac Pdf

Vertex Algebras introduces the algebraic aspects of conformal field theory and provides access to many unusual mathematical notions.