Homotopy Quantum Field Theory
Homotopy Quantum Field Theory

Author: Vladimir G. Turaev

Publisher: European Mathematical Society

Published: 2010

Total Pages: 300

ISBN-13: 9783037190869

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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.

Hqfts and Quantum Groups
Hqfts and Quantum Groups

Author: Neha Gupta

Publisher: LAP Lambert Academic Publishing

Published: 2012

Total Pages: 144

ISBN-13: 9783847338574

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The thesis is divided into two parts one for dimension 2 and the other for dimension 3. Part one of the thesis generalizes the de nition of an n-dimensional HQFT in terms of a monoidal functor from a rigid symmetric monoidal category X Cobn to any monoidal category A. In particular, 2-dimensional HQFTs with target K(G,1) taking values in A are generated from any Turaev G-crossed system in A and vice-versa. This is the generalization of Turaev's theory into a purely categorical set-up. Part two of the thesis generalizes the concept of a group-coalgebra, Hopf group-coalgebra, crossed Hopf group-coalgebra and quasitriangular Hopf group-coalgebra over a group scheme. Quantum double of a crossed Hopf group-scheme coalgebra is constructed in the a ne case and conjectured for non-a ne case. We can construct 3-dimensional HQFTs from modular crossed G-categories. The category of representations of a quantum double of a crossed Hopf group-coalgebra is a ribbon(quasitriangular) crossed group-category. Hence generates 3-dimensional HQFTs under certain conditions if the category becomes modular. However, systematic nding of modular crossed G-categories is largely open."

Quantum Groups, Quantum Categories and Quantum Field Theory
Quantum Groups, Quantum Categories and Quantum Field Theory

Author: Jürg Fröhlich

Publisher: Springer

Published: 2006-11-15

Total Pages: 438

ISBN-13: 3540476113

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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.

Introduction to Quantum Groups
Introduction to Quantum Groups

Author: Masud Chaichian

Publisher: World Scientific

Published: 1996

Total Pages: 362

ISBN-13: 9789810226237

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In the past decade there has been an extemely rapid growth in the interest and development of quantum group theory.This book provides students and researchers with a practical introduction to the principal ideas of quantum groups theory and its applications to quantum mechanical and modern field theory problems. It begins with a review of, and introduction to, the mathematical aspects of quantum deformation of classical groups, Lie algebras and related objects (algebras of functions on spaces, differential and integral calculi). In the subsequent chapters the richness of mathematical structure and power of the quantum deformation methods and non-commutative geometry is illustrated on the different examples starting from the simplest quantum mechanical system — harmonic oscillator and ending with actual problems of modern field theory, such as the attempts to construct lattice-like regularization consistent with space-time Poincaré symmetry and to incorporate Higgs fields in the general geometrical frame of gauge theories. Graduate students and researchers studying the problems of quantum field theory, particle physics and mathematical aspects of quantum symmetries will find the book of interest.

Introduction To Quantum Groups
Introduction To Quantum Groups

Author: Masud Chaichian

Publisher: World Scientific

Published: 1996-11-22

Total Pages: 357

ISBN-13: 9814499137

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In the past decade there has been an extemely rapid growth in the interest and development of quantum group theory.This book provides students and researchers with a practical introduction to the principal ideas of quantum groups theory and its applications to quantum mechanical and modern field theory problems. It begins with a review of, and introduction to, the mathematical aspects of quantum deformation of classical groups, Lie algebras and related objects (algebras of functions on spaces, differential and integral calculi). In the subsequent chapters the richness of mathematical structure and power of the quantum deformation methods and non-commutative geometry is illustrated on the different examples starting from the simplest quantum mechanical system — harmonic oscillator and ending with actual problems of modern field theory, such as the attempts to construct lattice-like regularization consistent with space-time Poincaré symmetry and to incorporate Higgs fields in the general geometrical frame of gauge theories. Graduate students and researchers studying the problems of quantum field theory, particle physics and mathematical aspects of quantum symmetries will find the book of interest.

Quantum Field Theory and Topology
Quantum Field Theory and Topology

Author: Albert S. Schwarz

Publisher: Springer Science & Business Media

Published: 2013-04-09

Total Pages: 277

ISBN-13: 366202943X

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In recent years topology has firmly established itself as an important part of the physicist's mathematical arsenal. It has many applications, first of all in quantum field theory, but increasingly also in other areas of physics. The main focus of this book is on the results of quantum field theory that are obtained by topological methods. Some aspects of the theory of condensed matter are also discussed. Part I is an introduction to quantum field theory: it discusses the basic Lagrangians used in the theory of elementary particles. Part II is devoted to the applications of topology to quantum field theory. Part III covers the necessary mathematical background in summary form. The book is aimed at physicists interested in applications of topology to physics and at mathematicians wishing to familiarize themselves with quantum field theory and the mathematical methods used in this field. It is accessible to graduate students in physics and mathematics.

Quantum Groups
Quantum Groups

Author: Petr P. Kulish

Publisher:

Published: 1992

Total Pages: 420

ISBN-13:

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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.

Introduction to Quantum Groups
Introduction to Quantum Groups

Author: Teo Banica

Publisher: Springer Nature

Published: 2023-01-01

Total Pages: 428

ISBN-13: 3031238176

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This book introduces the reader to quantum groups, focusing on the simplest ones, namely the closed subgroups of the free unitary group. Although such quantum groups are quite easy to understand mathematically, interesting examples abound, including all classical Lie groups, their free versions, half-liberations, other intermediate liberations, anticommutation twists, the duals of finitely generated discrete groups, quantum permutation groups, quantum reflection groups, quantum symmetry groups of finite graphs, and more. The book is written in textbook style, with its contents roughly covering a one-year graduate course. Besides exercises, the author has included many remarks, comments and pieces of advice with the lone reader in mind. The prerequisites are basic algebra, analysis and probability, and a certain familiarity with complex analysis and measure theory. Organized in four parts, the book begins with the foundations of the theory, due to Woronowicz, comprising axioms, Haar measure, Peter–Weyl theory, Tannakian duality and basic Brauer theorems. The core of the book, its second and third parts, focus on the main examples, first in the continuous case, and then in the discrete case. The fourth and last part is an introduction to selected research topics, such as toral subgroups, homogeneous spaces and matrix models. Introduction to Quantum Groups offers a compelling introduction to quantum groups, from the simplest examples to research level topics.