Mathematical Aspects Of Conformal And Topological Field Theories And Quantum Groups

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Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups

Paul J. Sally (Jr.)

Author : Paul J. Sally (Jr.)
Publisher : American Mathematical Soc.
Page : 282 pages
File Size : 49,8 Mb
Release : 1994
Category : Mathematics
ISBN : 9780821851869

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Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups by Paul J. Sally (Jr.) Pdf

This book contains papers presented by speakers at the AMS-IMS-SIAM Joint Summer Research Conference on Conformal Field Theory, Topological Field Theory and Quantum Groups, held at Mount Holyoke College in June 1992. One group of papers deals with one aspect of conformal field theory, namely, vertex operator algebras or superalgebras and their representations. Another group deals with various aspects of quantum groups. Other topics covered include the theory of knots in three-manifolds, symplectic geometry, and tensor products. This book provides an excellent view of some of the latest developments in this growing field of research.

Integrable Systems, Quantum Groups, and Quantum Field Theories

Alberto Ibort,M.A. Rodríguez

Author : Alberto Ibort,M.A. Rodríguez
Publisher : Springer Science & Business Media
Page : 508 pages
File Size : 54,8 Mb
Release : 2012-12-06
Category : Science
ISBN : 9789401119801

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Integrable Systems, Quantum Groups, and Quantum Field Theories by Alberto Ibort,M.A. Rodríguez Pdf

In many ways the last decade has witnessed a surge of interest in the interplay between theoretical physics and some traditional areas of pure mathematics. This book contains the lectures delivered at the NATO-ASI Summer School on `Recent Problems in Mathematical Physics' held at Salamanca, Spain (1992), offering a pedagogical and updated approach to some of the problems that have been at the heart of these events. Among them, we should mention the new mathematical structures related to integrability and quantum field theories, such as quantum groups, conformal field theories, integrable statistical models, and topological quantum field theories, that are discussed at length by some of the leading experts on the areas in several of the lectures contained in the book. Apart from these, traditional and new problems in quantum gravity are reviewed. Other contributions to the School included in the book range from symmetries in partial differential equations to geometrical phases in quantum physics. The book is addressed to researchers in the fields covered, PhD students and any scientist interested in obtaining an updated view of the subjects.

Mathematical Aspects of Quantum Field Theories

Damien Calaque,Thomas Strobl

Author : Damien Calaque,Thomas Strobl
Publisher : Springer
Page : 572 pages
File Size : 42,7 Mb
Release : 2015-01-06
Category : Science
ISBN : 9783319099491

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Mathematical Aspects of Quantum Field Theories by Damien Calaque,Thomas Strobl Pdf

Despite its long history and stunning experimental successes, the mathematical foundation of perturbative quantum field theory is still a subject of ongoing research. This book aims at presenting some of the most recent advances in the field, and at reflecting the diversity of approaches and tools invented and currently employed. Both leading experts and comparative newcomers to the field present their latest findings, helping readers to gain a better understanding of not only quantum but also classical field theories. Though the book offers a valuable resource for mathematicians and physicists alike, the focus is more on mathematical developments. This volume consists of four parts: The first Part covers local aspects of perturbative quantum field theory, with an emphasis on the axiomatization of the algebra behind the operator product expansion. The second Part highlights Chern-Simons gauge theories, while the third examines (semi-)classical field theories. In closing, Part 4 addresses factorization homology and factorization algebras.

Quantum Groups, Quantum Categories and Quantum Field Theory

Jürg Fröhlich,Thomas Kerler

Author : Jürg Fröhlich,Thomas Kerler
Publisher : Springer
Page : 438 pages
File Size : 50,6 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540476115

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Quantum Groups, Quantum Categories and Quantum Field Theory by Jürg Fröhlich,Thomas Kerler Pdf

This book reviews recent results on low-dimensional quantum field theories and their connection with quantum group theory and the theory of braided, balanced tensor categories. It presents detailed, mathematically precise introductions to these subjects and then continues with new results. Among the main results are a detailed analysis of the representation theory of U (sl ), for q a primitive root of unity, and a semi-simple quotient thereof, a classfication of braided tensor categories generated by an object of q-dimension less than two, and an application of these results to the theory of sectors in algebraic quantum field theory. This clarifies the notion of "quantized symmetries" in quantum fieldtheory. The reader is expected to be familiar with basic notions and resultsin algebra. The book is intended for research mathematicians, mathematical physicists and graduate students.

Quantum Topology

Louis H Kauffman,Randy A Baadhio

Author : Louis H Kauffman,Randy A Baadhio
Publisher : World Scientific
Page : 392 pages
File Size : 55,7 Mb
Release : 1993-09-15
Category : Science
ISBN : 9789814502672

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Quantum Topology by Louis H Kauffman,Randy A Baadhio Pdf

This book constitutes a review volume on the relatively new subject of Quantum Topology. Quantum Topology has its inception in the 1984/1985 discoveries of new invariants of knots and links (Jones, Homfly and Kauffman polynomials). These invariants were rapidly connected with quantum groups and methods in statistical mechanics. This was followed by Edward Witten's introduction of methods of quantum field theory into the subject and the formulation by Witten and Michael Atiyah of the concept of topological quantum field theories. This book is a review volume of on-going research activity. The papers derive from talks given at the Special Session on Knot and Topological Quantum Field Theory of the American Mathematical Society held at Dayton, Ohio in the fall of 1992. The book consists of a self-contained article by Kauffman, entitled Introduction to Quantum Topology and eighteen research articles by participants in the special session. This book should provide a useful source of ideas and results for anyone interested in the interface between topology and quantum field theory. Contents:Introduction to Quantum Topology (L H Kauffman)Knot Theory, Exotic Spheres and Global Gravitational Anomalies (R A Baadhio)A Diagrammatic Theory of Knotted Surfaces (J S Carter & M Saito)A Categorical Construction of 4D Topological Quantum Field Theories (L Crane & D Yetter)Evaluating the Crane-Yetter Invariant (L Crane, L H Kauffman & D Yetter)A Method for Computing the Arf Invariants of Links (P Gilmer)Triangulations, Categories and Extended Topological Field Theories (R J Lawrence)The Casson Invariant for Two-Fold Branched Covers of Links (D Mullins)Elementary Conjectures in Classical Knot Theory (J H Przytycki)Knot Polynomials as States of Nonperturbative Four Dimensional Quantum Gravity (J Pullin)On Invariants of 3-Manifolds Derived from Abelian Groups (J Mattes, M M Polyak & N Reshetikhin)and other papers Readership: Mathematicians and mathematical physicists. keywords:Quantum Topology;Topological Quantum Field Theory;Meeting;AMS Special Session;Dayton, OH (USA)

Geometric and Topological Methods for Quantum Field Theory

Alexander Cardona,Sylvie Paycha,Hernan Ocampo

Author : Alexander Cardona,Sylvie Paycha,Hernan Ocampo
Publisher : World Scientific
Page : 492 pages
File Size : 44,5 Mb
Release : 2003-03-21
Category : Mathematics
ISBN : 9789814487672

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Geometric and Topological Methods for Quantum Field Theory by Alexander Cardona,Sylvie Paycha,Hernan Ocampo Pdf

This volume offers an introduction to recent developments in several active topics of research at the interface between geometry, topology and quantum field theory. These include Hopf algebras underlying renormalization schemes in quantum field theory, noncommutative geometry with applications to index theory on one hand and the study of aperiodic solids on the other, geometry and topology of low dimensional manifolds with applications to topological field theory, Chern-Simons supergravity and the anti de Sitter/conformal field theory correspondence. It comprises seven lectures organized around three main topics, noncommutative geometry, topological field theory, followed by supergravity and string theory, complemented by some short communications by young participants of the school. Contents:Noncommutative Geometry:Hopf Algebras in Noncommutative Geometry (J C Várilly)The Noncommutative Geometry of Aperiodic Solids (J Bellissard)Noncommutative Geometry and Abstract Integration Theory (M-T Benameur)Topological Field Theory:Introduction to Quantum Invariants of 3-Manifolds, Topological Quantum Field Theories and Modular Categories (C Blanchet)An Introduction to Donaldson–Witten Theory (M Mariño)Supergravity and String Theory:(Super)-Gravities Beyond 4 Dimensions (J Zanelli)Introductory Lectures on String Theory and the AdS/CFT Correspondence (A Pankiewicz & S Theisen)Short Communications:Group Contractions and Its Consequences Upon Representations of Different Spatial Symmetry Groups (M Ayala-Sánchez & R W Haase)Phase Anomalies as Trace Anomalies in Chern–Simons Theory (A Cardona)Deligne Cohomology for Orbifolds, Discrete Torsion and B-Fields (E Lupercio & B Uribe) Readership: Graduate students and researchers in theoretical and mathematical physics, as well as geometry and topology. Keywords:

Quantum Invariants of Knots and 3-Manifolds

Vladimir G. Turaev

Author : Vladimir G. Turaev
Publisher : Walter de Gruyter GmbH & Co KG
Page : 600 pages
File Size : 53,8 Mb
Release : 2020-03-23
Category : Mathematics
ISBN : 9783110883275

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Quantum Invariants of Knots and 3-Manifolds by Vladimir G. Turaev Pdf

This monograph provides a systematic treatment of topological quantum field theories (TQFT's) in three dimensions, inspired by the discovery of the Jones polynomial of knots, the Witten-Chern-Simons field theory, and the theory of quantum groups. The author, one of the leading experts in the subject, gives a rigorous and self-contained exposition of new fundamental algebraic and topological concepts that emerged in this theory. The book is divided into three parts. Part I presents a construction of 3-dimensional TQFT's and 2-dimensional modular functors from so-called modular categories. This gives new knot and 3-manifold invariants as well as linear representations of the mapping class groups of surfaces. In Part II the machinery of 6j-symbols is used to define state sum invariants of 3-manifolds. Their relation to the TQFT's constructed in Part I is established via the theory of shadows. Part III provides constructions of modular categories, based on quantum groups and Kauffman's skein modules. This book is accessible to graduate students in mathematics and physics with a knowledge of basic algebra and topology. It will be an indispensable source for everyone who wishes to enter the forefront of this rapidly growing and fascinating area at the borderline of mathematics and physics. Most of the results and techniques presented here appear in book form for the first time.

Quantum Field Theory Conformal Group Theory Conformal Field Theory

R. Mirman

Author : R. Mirman
Publisher : iUniverse
Page : 313 pages
File Size : 52,5 Mb
Release : 2005-02
Category : Science
ISBN : 9780595336920

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Quantum Field Theory Conformal Group Theory Conformal Field Theory by R. Mirman Pdf

The conformal group is the invariance group of geometry (which is not understood), the largest one. Physical applications are implied, as discussed, including reasons for interactions. The group structure as well as those of related groups are analyzed. An inhomogeneous group is a subgroup of a homogeneous one because of nonlinearities of the realization. Conservation of baryons (protons can't decay) is explained and proven. Reasons for various realizations, so matrix elements, of the Lorentz group given. The clearly relevant mass level formula is compared with experimental values. Questions, implications and possibilities, including for differential equations, are raised.

Quantum Topology - Proceedings Of The Conference

David N Yetter

Author : David N Yetter
Publisher : World Scientific
Page : 390 pages
File Size : 44,9 Mb
Release : 1994-08-19
Category : Electronic
ISBN : 9789814551595

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Quantum Topology - Proceedings Of The Conference by David N Yetter Pdf

This volume contains the conference on quantum topology, held at Kansas State University, Manhattan, KS, 24 - 28 March 1993.Quantum topology is a rapidly growing field of mathematics dealing with the recently discovered interactions between low-dimensional topology, the theory of quantum groups, category theory, C∗-algebra theory, gauge theory, conformal and topological field theory and statistical mechanics. The conference, attended by over 60 mathematicians and theoretical physicists from Canada, Denmark, England, France, Japan, Poland and the United States, was highlighted by lecture series given by Louis Kauffman, Univ. of Illinois at Chicago and Nicholai Reshetikhin, Univ. of Califonia, Berkeley.

Quantum Groups

Petr P. Kulish

Author : Petr P. Kulish
Publisher : Unknown
Page : 424 pages
File Size : 45,5 Mb
Release : 1992
Category : Mathematics
ISBN : UOM:39015028426354

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Quantum Groups by Petr P. Kulish Pdf

The theory of Quantum Groups is a rapidly developing area with numerous applications in mathematics and theoretical physics, e.g. in link and knot invariants in topology, q-special functions, conformal field theory, quantum integrable models. The aim of the Euler Institute's workshops was to review and compile the progress achieved in the different subfields. Near 100 participants came from 14 countries. More than 20 contributions written up for this book contain new, unpublished material and half of them include a survey of recent results in the field (deformation theory, graded differential algebras, contraction technique, knot invariants, q-special functions). FROM THE CONTENTS: V.G. Drinfeld: On Some Unsolved Problems in Quantum Group Theory.- M. Gerstenhaber, A. Giaquinto, S.D. Schack: Quantum Symmetry.- L.I. Korogodsky, L.L. Vaksman: Quantum G-Spaces and Heisenberg Algebra.-J. Stasheff: Differential Graded Lie Algebras, Quasi-Hopf Algebras and Higher Homotopy Algebras.- A. Yu. Alekseev, L.D. Faddeev, M.A. Semenov-Tian-Shansky: Hidden Quantum Groups inside Kac-Moody Algebras.- J.-L. Gervais: Quantum Group Symmetry of 2D Gravity.- T. Kohno: Invariants of 3-Manifolds Based on Conformal Field Theory and Heegaard Splitting.- O. Viro: Moves of Triangulations of a PL-Manifold.-- Publisher description.

Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners

Thomas Kerler,Volodymyr V. Lyubashenko

Author : Thomas Kerler,Volodymyr V. Lyubashenko
Publisher : Springer
Page : 383 pages
File Size : 54,6 Mb
Release : 2003-07-01
Category : Mathematics
ISBN : 9783540446255

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Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners by Thomas Kerler,Volodymyr V. Lyubashenko Pdf

This book presents the (to date) most general approach to combinatorial constructions of topological quantum field theories (TQFTs) in three dimensions. The authors describe extended TQFTs as double functors between two naturally defined double categories: one of topological nature, made of 3-manifolds with corners, the other of algebraic nature, made of linear categories, functors, vector spaces and maps. Atiyah's conventional notion of TQFTs as well as the notion of modular functor from axiomatic conformal field theory are unified in this concept. A large class of such extended modular catergory is constructed, assigning a double functor to every abelian modular category, which does not have to be semisimple.

Homotopy Quantum Field Theory

Vladimir G. Turaev

Author : Vladimir G. Turaev
Publisher : European Mathematical Society
Page : 300 pages
File Size : 44,6 Mb
Release : 2010
Category : EMS tracts in mathematics
ISBN : 3037190868

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Homotopy Quantum Field Theory by Vladimir G. Turaev Pdf

Homotopy Quantum Field Theory (HQFT) is a branch of Topological Quantum Field Theory founded by E. Witten and M. Atiyah. It applies ideas from theoretical physics to study principal bundles over manifolds and, more generally, homotopy classes of maps from manifolds to a fixed target space. This book is the first systematic exposition of Homotopy Quantum Field Theory. It starts with a formal definition of an HQFT and provides examples of HQFTs in all dimensions. The main body of the text is focused on $2$-dimensional and $3$-dimensional HQFTs. A study of these HQFTs leads to new algebraic objects: crossed Frobenius group-algebras, crossed ribbon group-categories, and Hopf group-coalgebras. These notions and their connections with HQFTs are discussed in detail. The text ends with several appendices including an outline of recent developments and a list of open problems. Three appendices by M. Muger and A. Virelizier summarize their work in this area. The book is addressed to mathematicians, theoretical physicists, and graduate students interested in topological aspects of quantum field theory. The exposition is self-contained and well suited for a one-semester graduate course. Prerequisites include only basics of algebra and topology.

Conformal Field Theories and Tensor Categories

Chengming Bai,Jürgen Fuchs,Yi-Zhi Huang,Liang Kong,Ingo Runkel,Christoph Schweigert

Author : Chengming Bai,Jürgen Fuchs,Yi-Zhi Huang,Liang Kong,Ingo Runkel,Christoph Schweigert
Publisher : Springer Science & Business Media
Page : 285 pages
File Size : 47,6 Mb
Release : 2013-10-30
Category : Mathematics
ISBN : 9783642393839

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Conformal Field Theories and Tensor Categories by Chengming Bai,Jürgen Fuchs,Yi-Zhi Huang,Liang Kong,Ingo Runkel,Christoph Schweigert Pdf

The present volume is a collection of seven papers that are either based on the talks presented at the workshop "Conformal field theories and tensor categories" held June 13 to June 17, 2011 at the Beijing International Center for Mathematical Research, Peking University, or are extensions of the material presented in the talks at the workshop. These papers present new developments beyond rational conformal field theories and modular tensor categories and new applications in mathematics and physics. The topics covered include tensor categories from representation categories of Hopf algebras, applications of conformal field theories and tensor categories to topological phases and gapped systems, logarithmic conformal field theories and the corresponding non-semisimple tensor categories, and new developments in the representation theory of vertex operator algebras. Some of the papers contain detailed introductory material that is helpful for graduate students and researchers looking for an introduction to these research directions. The papers also discuss exciting recent developments in the area of conformal field theories, tensor categories and their applications and will be extremely useful for researchers working in these areas.

Non-perturbative Quantum Field Theory: Mathematical Aspects And Applications

Jurg Frohlich

Author : Jurg Frohlich
Publisher : World Scientific
Page : 854 pages
File Size : 47,6 Mb
Release : 1992-04-29
Category : Electronic
ISBN : 9789814506564

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Non-perturbative Quantum Field Theory: Mathematical Aspects And Applications by Jurg Frohlich Pdf

Compiled to illustrate the recent history of Quantum Field Theory and its trends, this collection of selected reprints by Jürg Fröhlich, a leading theoretician in the field, is a comprehensive guide of the more mathematical aspects of the subject. Results and methods of the past fifteen years are reviewed. The analytical methods employed are non-perturbative and, for the larger part, mathematically rigorous. Most articles are review articles surveying certain important developments in quantum field theory and guiding the reader towards the original literature.The volume begins with a comprehensive introduction by Jürg Fröhlich.The theory of phase transitions and continuous symmetry breaking is reviewed in the first section. The second section discusses the non-perturbative quantization of topological solitons. The third section is devoted to the study of gauge fields. A paper on the triviality of λϖ4 — theory in four and more dimensions is found in the fourth section, while the fifth contains two articles on “random geometry”. The sixth and final part addresses topics in low-dimensional quantum field theory, including braid statistics, two-dimensional conformal field theory and an application to condensed matter theory.

Boundary Conformal Field Theory and the Worldsheet Approach to D-Branes

Andreas Recknagel,Volker Schomerus

Author : Andreas Recknagel,Volker Schomerus
Publisher : Cambridge University Press
Page : 349 pages
File Size : 51,5 Mb
Release : 2013-11-28
Category : Mathematics
ISBN : 9780521832236

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Boundary Conformal Field Theory and the Worldsheet Approach to D-Branes by Andreas Recknagel,Volker Schomerus Pdf

A comprehensive introduction to the mathematical description of strings, D-branes and the geometry of strongly curved spacetime.

Mathematical Aspects Of Conformal And Topological Field Theories And Quantum Groups

Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups by Paul J. Sally (Jr.) Pdf

Integrable Systems, Quantum Groups, and Quantum Field Theories by Alberto Ibort,M.A. Rodríguez Pdf

Mathematical Aspects of Quantum Field Theories by Damien Calaque,Thomas Strobl Pdf

Quantum Groups, Quantum Categories and Quantum Field Theory by Jürg Fröhlich,Thomas Kerler Pdf

Quantum Topology by Louis H Kauffman,Randy A Baadhio Pdf

Geometric and Topological Methods for Quantum Field Theory by Alexander Cardona,Sylvie Paycha,Hernan Ocampo Pdf

Quantum Invariants of Knots and 3-Manifolds by Vladimir G. Turaev Pdf

Quantum Field Theory Conformal Group Theory Conformal Field Theory by R. Mirman Pdf

Quantum Topology - Proceedings Of The Conference by David N Yetter Pdf

Quantum Groups by Petr P. Kulish Pdf

Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners by Thomas Kerler,Volodymyr V. Lyubashenko Pdf

Homotopy Quantum Field Theory by Vladimir G. Turaev Pdf

Conformal Field Theories and Tensor Categories by Chengming Bai,Jürgen Fuchs,Yi-Zhi Huang,Liang Kong,Ingo Runkel,Christoph Schweigert Pdf

Non-perturbative Quantum Field Theory: Mathematical Aspects And Applications by Jurg Frohlich Pdf

Boundary Conformal Field Theory and the Worldsheet Approach to D-Branes by Andreas Recknagel,Volker Schomerus Pdf

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