Monoidal Categories And Topological Field Theory

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Monoidal Categories and Topological Field Theory

Vladimir Turaev,Alexis Virelizier
Monoidal Categories and Topological Field Theory Book PDF

Author : Vladimir Turaev,Alexis Virelizier
Publisher : Unknown
Page : 523 pages
File Size : 54,7 Mb
Release : 2017
Category : Algebra, Homological
ISBN : 3319498355

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Monoidal Categories and Topological Field Theory by Vladimir Turaev,Alexis Virelizier Pdf

This monograph is devoted to monoidal categories and their connections with 3-dimensional topological field theories. Starting with basic definitions, it proceeds to the forefront of current research. Part 1 introduces monoidal categories and several of their classes, including rigid, pivotal, spherical, fusion, braided, and modular categories. It then presents deep theorems of Müger on the center of a pivotal fusion category. These theorems are proved in Part 2 using the theory of Hopf monads. In Part 3 the authors define the notion of a topological quantum field theory (TQFT) and construct a Turaev-Viro-type 3-dimensional state sum TQFT from a spherical fusion category. Lastly, in Part 4 this construction is extended to 3-manifolds with colored ribbon graphs, yielding a so-called graph TQFT (and, consequently, a 3-2-1 extended TQFT). The authors then prove the main result of the monograph: the state sum graph TQFT derived from any spherical fusion category is isomorphic to the Reshetikhin-Turaev surgery graph TQFT derived from the center of that category. The book is of interest to researchers and students studying topological field theory, monoidal categories, Hopf algebras and Hopf monads.

Monoidal Categories and Topological Field Theory

Vladimir Turaev,Alexis Virelizier
Monoidal Categories and Topological Field Theory Book PDF

Author : Vladimir Turaev,Alexis Virelizier
Publisher : Birkhäuser
Page : 523 pages
File Size : 48,9 Mb
Release : 2017-06-28
Category : Mathematics
ISBN : 9783319498348

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Monoidal Categories and Topological Field Theory by Vladimir Turaev,Alexis Virelizier Pdf

This monograph is devoted to monoidal categories and their connections with 3-dimensional topological field theories. Starting with basic definitions, it proceeds to the forefront of current research. Part 1 introduces monoidal categories and several of their classes, including rigid, pivotal, spherical, fusion, braided, and modular categories. It then presents deep theorems of Müger on the center of a pivotal fusion category. These theorems are proved in Part 2 using the theory of Hopf monads. In Part 3 the authors define the notion of a topological quantum field theory (TQFT) and construct a Turaev-Viro-type 3-dimensional state sum TQFT from a spherical fusion category. Lastly, in Part 4 this construction is extended to 3-manifolds with colored ribbon graphs, yielding a so-called graph TQFT (and, consequently, a 3-2-1 extended TQFT). The authors then prove the main result of the monograph: the state sum graph TQFT derived from any spherical fusion category is isomorphic to the Reshetikhin-Turaev surgery graph TQFT derived from the center of that category. The book is of interest to researchers and students studying topological field theory, monoidal categories, Hopf algebras and Hopf monads.

Frobenius Algebras and 2-D Topological Quantum Field Theories

Joachim Kock
Frobenius Algebras and 2 D Topological Quantum Field Theories Book PDF

Author : Joachim Kock
Publisher : Cambridge University Press
Page : 260 pages
File Size : 49,8 Mb
Release : 2004
Category : Mathematics
ISBN : 0521540313

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Frobenius Algebras and 2-D Topological Quantum Field Theories by Joachim Kock Pdf

This 2003 book describes a striking connection between topology and algebra, namely that 2D topological quantum field theories are equivalent to commutative Frobenius algebras. The precise formulation of the theorem and its proof is given in terms of monoidal categories, and the main purpose of the book is to develop these concepts from an elementary level, and more generally serve as an introduction to categorical viewpoints in mathematics. Rather than just proving the theorem, it is shown how the result fits into a more general pattern concerning universal monoidal categories for algebraic structures. Throughout, the emphasis is on the interplay between algebra and topology, with graphical interpretation of algebraic operations, and topological structures described algebraically in terms of generators and relations. The book will prove valuable to students or researchers entering this field who will learn a host of modern techniques that will prove useful for future work.

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

Thomas Kerler,Volodymyr V. Lyubashenko
Non Semisimple Topological Quantum Field Theories for 3 Manifolds with Corners Book PDF

Author : Thomas Kerler,Volodymyr V. Lyubashenko
Publisher : Springer
Page : 383 pages
File Size : 47,5 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.

Lectures on Factorization Homology, ∞-Categories, and Topological Field Theories

Hiro Lee Tanaka
Lectures on Factorization Homology Categories and Topological Field Theories Book PDF

Author : Hiro Lee Tanaka
Publisher : Springer Nature
Page : 84 pages
File Size : 42,5 Mb
Release : 2020-12-14
Category : Science
ISBN : 9783030611637

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Lectures on Factorization Homology, ∞-Categories, and Topological Field Theories by Hiro Lee Tanaka Pdf

This book provides an informal and geodesic introduction to factorization homology, focusing on providing intuition through simple examples. Along the way, the reader is also introduced to modern ideas in homotopy theory and category theory, particularly as it relates to the use of infinity-categories. As with the original lectures, the text is meant to be a leisurely read suitable for advanced graduate students and interested researchers in topology and adjacent fields.

Lectures on Field Theory and Topology

Daniel S. Freed
Lectures on Field Theory and Topology Book PDF

Author : Daniel S. Freed
Publisher : American Mathematical Soc.
Page : 186 pages
File Size : 50,6 Mb
Release : 2019-08-23
Category : Algebraic topology
ISBN : 9781470452063

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Lectures on Field Theory and Topology by Daniel S. Freed Pdf

These lectures recount an application of stable homotopy theory to a concrete problem in low energy physics: the classification of special phases of matter. While the joint work of the author and Michael Hopkins is a focal point, a general geometric frame of reference on quantum field theory is emphasized. Early lectures describe the geometric axiom systems introduced by Graeme Segal and Michael Atiyah in the late 1980s, as well as subsequent extensions. This material provides an entry point for mathematicians to delve into quantum field theory. Classification theorems in low dimensions are proved to illustrate the framework. The later lectures turn to more specialized topics in field theory, including the relationship between invertible field theories and stable homotopy theory, extended unitarity, anomalies, and relativistic free fermion systems. The accompanying mathematical explanations touch upon (higher) category theory, duals to the sphere spectrum, equivariant spectra, differential cohomology, and Dirac operators. The outcome of computations made using the Adams spectral sequence is presented and compared to results in the condensed matter literature obtained by very different means. The general perspectives and specific applications fuse into a compelling story at the interface of contemporary mathematics and theoretical physics.

Homotopy Quantum Field Theory

Vladimir G. Turaev
Homotopy Quantum Field Theory Book PDF

Author : Vladimir G. Turaev
Publisher : European Mathematical Society
Page : 300 pages
File Size : 50,8 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.

Quantum Invariants of Knots and 3-Manifolds

Vladimir G. Turaev
Quantum Invariants of Knots and 3 Manifolds Book PDF

Author : Vladimir G. Turaev
Publisher : Walter de Gruyter GmbH & Co KG
Page : 600 pages
File Size : 45,5 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.

Topology and Field Theories

Stephan Stolz
Topology and Field Theories Book PDF

Author : Stephan Stolz
Publisher : American Mathematical Soc.
Page : 186 pages
File Size : 50,9 Mb
Release : 2014-04-17
Category : Mathematics
ISBN : 9781470410155

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Topology and Field Theories by Stephan Stolz Pdf

This book is a collection of expository articles based on four lecture series presented during the 2012 Notre Dame Summer School in Topology and Field Theories. The four topics covered in this volume are: Construction of a local conformal field theory associated to a compact Lie group, a level and a Frobenius object in the corresponding fusion category; Field theory interpretation of certain polynomial invariants associated to knots and links; Homotopy theoretic construction of far-reaching generalizations of the topological field theories that Dijkgraf and Witten associated to finite groups; and a discussion of the action of the orthogonal group on the full subcategory of an -category consisting of the fully dualizable objects. The expository style of the articles enables non-experts to understand the basic ideas of this wide range of important topics.

Quantum Groups, Quantum Categories and Quantum Field Theory

Jürg Fröhlich,Thomas Kerler
Quantum Groups Quantum Categories and Quantum Field Theory Book PDF

Author : Jürg Fröhlich,Thomas Kerler
Publisher : Springer
Page : 438 pages
File Size : 49,9 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.

Conformal Field Theories and Tensor Categories

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

Author : Chengming Bai,Jürgen Fuchs,Yi-Zhi Huang,Liang Kong,Ingo Runkel,Christoph Schweigert
Publisher : Springer Science & Business Media
Page : 285 pages
File Size : 46,6 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.

Quantum Field Theory and Manifold Invariants

Daniel S. Freed,Sergei Gukov,Ciprian Manolescu,Constantin Teleman,Ulrike Tillmann
Quantum Field Theory and Manifold Invariants Book PDF

Author : Daniel S. Freed,Sergei Gukov,Ciprian Manolescu,Constantin Teleman,Ulrike Tillmann
Publisher : American Mathematical Society, IAS/Park City Mathematics Institute
Page : 476 pages
File Size : 53,5 Mb
Release : 2021-12-02
Category : Mathematics
ISBN : 9781470461232

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Quantum Field Theory and Manifold Invariants by Daniel S. Freed,Sergei Gukov,Ciprian Manolescu,Constantin Teleman,Ulrike Tillmann Pdf

This volume contains lectures from the Graduate Summer School “Quantum Field Theory and Manifold Invariants” held at Park City Mathematics Institute 2019. The lectures span topics in topology, global analysis, and physics, and they range from introductory to cutting edge. Topics treated include mathematical gauge theory (anti-self-dual equations, Seiberg-Witten equations, Higgs bundles), classical and categorified knot invariants (Khovanov homology, Heegaard Floer homology), instanton Floer homology, invertible topological field theory, BPS states and spectral networks. This collection presents a rich blend of geometry and topology, with some theoretical physics thrown in as well, and so provides a snapshot of a vibrant and fast-moving field. Graduate students with basic preparation in topology and geometry can use this volume to learn advanced background material before being brought to the frontiers of current developments. Seasoned researchers will also benefit from the systematic presentation of exciting new advances by leaders in their fields.

Functorial Knot Theory

David N. Yetter
Functorial Knot Theory Book PDF

Author : David N. Yetter
Publisher : World Scientific
Page : 238 pages
File Size : 44,6 Mb
Release : 2001
Category : Mathematics
ISBN : 9789810244439

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Functorial Knot Theory by David N. Yetter Pdf

Almost since the advent of skein-theoretic invariants of knots and links (the Jones, HOMFLY, and Kauffman polynomials), the important role of categories of tangles in the connection between low-dimensional topology and quantum-group theory has been recognized. The rich categorical structures naturally arising from the considerations of cobordisms have suggested functorial views of topological field theory.This book begins with a detailed exposition of the key ideas in the discovery of monoidal categories of tangles as central objects of study in low-dimensional topology. The focus then turns to the deformation theory of monoidal categories and the related deformation theory of monoidal functors, which is a proper generalization of Gerstenhaber's deformation theory of associative algebras. These serve as the building blocks for a deformation theory of braided monoidal categories which gives rise to sequences of Vassiliev invariants of framed links, and clarify their interrelations.

Mathematical Foundations of Quantum Field Theory and Perturbative String Theory

Hisham Sati,Urs Schreiber
Mathematical Foundations of Quantum Field Theory and Perturbative String Theory Book PDF

Author : Hisham Sati,Urs Schreiber
Publisher : American Mathematical Soc.
Page : 370 pages
File Size : 47,8 Mb
Release : 2011-12-07
Category : Mathematics
ISBN : 9780821851951

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Mathematical Foundations of Quantum Field Theory and Perturbative String Theory by Hisham Sati,Urs Schreiber Pdf

Conceptual progress in fundamental theoretical physics is linked with the search for the suitable mathematical structures that model the physical systems. Quantum field theory (QFT) has proven to be a rich source of ideas for mathematics for a long time. However, fundamental questions such as ``What is a QFT?'' did not have satisfactory mathematical answers, especially on spaces with arbitrary topology, fundamental for the formulation of perturbative string theory. This book contains a collection of papers highlighting the mathematical foundations of QFT and its relevance to perturbative string theory as well as the deep techniques that have been emerging in the last few years. The papers are organized under three main chapters: Foundations for Quantum Field Theory, Quantization of Field Theories, and Two-Dimensional Quantum Field Theories. An introduction, written by the editors, provides an overview of the main underlying themes that bind together the papers in the volume.

Towards the Mathematics of Quantum Field Theory

Frédéric Paugam
Towards the Mathematics of Quantum Field Theory Book PDF

Author : Frédéric Paugam
Publisher : Springer Science & Business Media
Page : 485 pages
File Size : 49,7 Mb
Release : 2014-02-20
Category : Science
ISBN : 9783319045641

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Towards the Mathematics of Quantum Field Theory by Frédéric Paugam Pdf

This ambitious and original book sets out to introduce to mathematicians (even including graduate students ) the mathematical methods of theoretical and experimental quantum field theory, with an emphasis on coordinate-free presentations of the mathematical objects in use. This in turn promotes the interaction between mathematicians and physicists by supplying a common and flexible language for the good of both communities, though mathematicians are the primary target. This reference work provides a coherent and complete mathematical toolbox for classical and quantum field theory, based on categorical and homotopical methods, representing an original contribution to the literature. The first part of the book introduces the mathematical methods needed to work with the physicists' spaces of fields, including parameterized and functional differential geometry, functorial analysis, and the homotopical geometric theory of non-linear partial differential equations, with applications to general gauge theories. The second part presents a large family of examples of classical field theories, both from experimental and theoretical physics, while the third part provides an introduction to quantum field theory, presents various renormalization methods, and discusses the quantization of factorization algebras.