Topology And Field Theories

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Lectures on Field Theory and Topology

Author : Daniel S. Freed
Publisher : American Mathematical Soc.
Page : 186 pages
File Size : 43,6 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.

Quantum Field Theory and Topology

Author : Albert S. Schwarz
Publisher : Springer Science & Business Media
Page : 277 pages
File Size : 51,6 Mb
Release : 2013-04-09
Category : Mathematics
ISBN : 9783662029435

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Quantum Field Theory and Topology by Albert S. Schwarz Pdf

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.

Conformal Field Theory and Topology

Author : Toshitake Kohno
Publisher : American Mathematical Soc.
Page : 188 pages
File Size : 52,5 Mb
Release : 2002
Category : Mathematics
ISBN : 082182130X

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Conformal Field Theory and Topology by Toshitake Kohno Pdf

Geometry and physics have been developed with a strong influence on each other. One of the most remarkable interactions between geometry and physics since 1980 has been an application of quantum field theory to topology and differential geometry. This book focuses on a relationship between two-dimensional quantum field theory and three-dimensional topology which has been studied intensively since the discovery of the Jones polynomial in the middle of the 1980s and Witten's invariantfor 3-manifolds derived from Chern-Simons gauge theory. An essential difficulty in quantum field theory comes from infinite-dimensional freedom of a system. Techniques dealing with such infinite-dimensional objects developed in the framework of quantum field theory have been influential in geometryas well. This book gives an accessible treatment for a rigorous construction of topological invariants originally defined as partition functions of fields on manifolds. The book is organized as follows: The Introduction starts from classical mechanics and explains basic background materials in quantum field theory and geometry. Chapter 1 presents conformal field theory based on the geometry of loop groups. Chapter 2 deals with the holonomy of conformal field theory. Chapter 3 treatsChern-Simons perturbation theory. The final chapter discusses topological invariants for 3-manifolds derived from Chern-Simons perturbation theory.

Monoidal Categories and Topological Field Theory

Author : Vladimir Turaev,Alexis Virelizier
Publisher : Birkhäuser
Page : 523 pages
File Size : 43,5 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.

Topological Quantum Field Theory and Four Manifolds

Author : Jose Labastida,Marcos Marino
Publisher : Springer Science & Business Media
Page : 235 pages
File Size : 48,7 Mb
Release : 2007-07-18
Category : Science
ISBN : 9781402031779

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Topological Quantum Field Theory and Four Manifolds by Jose Labastida,Marcos Marino Pdf

The emergence of topological quantum ?eld theory has been one of the most important breakthroughs which have occurred in the context of ma- ematical physics in the last century, a century characterizedbyindependent developments of the main ideas in both disciplines, physics and mathematics, which has concluded with two decades of strong interaction between them, where physics, as in previous centuries, has acted as a source of new mat- matics. Topological quantum ?eld theories constitute the core of these p- nomena, although the main drivingforce behind it has been the enormous e?ort made in theoretical particle physics to understand string theory as a theory able to unify the four fundamental interactions observed in nature. These theories set up a new realm where both disciplines pro?t from each other. Although the most striking results have appeared on the mathema- calside,theoreticalphysicshasclearlyalsobene?tted,sincethecorresponding developments have helped better to understand aspects of the fundamentals of ?eld and string theory.

Differential Topology and Quantum Field Theory

Author : Charles Nash
Publisher : Elsevier
Page : 404 pages
File Size : 45,6 Mb
Release : 1991
Category : Mathematics
ISBN : 0125140762

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Differential Topology and Quantum Field Theory by Charles Nash Pdf

The remarkable developments in differential topology and how these recent advances have been applied as a primary research tool in quantum field theory are presented here in a style reflecting the genuinely two-sided interaction between mathematical physics and applied mathematics. The author, following his previous work (Nash/Sen: Differential Topology for Physicists, Academic Press, 1983), covers elliptic differential and pseudo-differential operators, Atiyah-Singer index theory, topological quantum field theory, string theory, and knot theory. The explanatory approach serves to illuminate and clarify these theories for graduate students and research workers entering the field for the first time. Treats differential geometry, differential topology, and quantum field theory Includes elliptic differential and pseudo-differential operators, Atiyah-Singer index theory, topological quantum field theory, string theory, and knot theory Tackles problems of quantum field theory using differential topology as a tool

Geometric and Topological Methods for Quantum Field Theory

Author : Hernan Ocampo,Eddy Pariguan,Sylvie Paycha
Publisher : Cambridge University Press
Page : 435 pages
File Size : 43,8 Mb
Release : 2010-04-29
Category : Science
ISBN : 9781139486736

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Geometric and Topological Methods for Quantum Field Theory by Hernan Ocampo,Eddy Pariguan,Sylvie Paycha Pdf

Aimed at graduate students in physics and mathematics, this book provides an introduction to recent developments in several active topics at the interface between algebra, geometry, topology and quantum field theory. The first part of the book begins with an account of important results in geometric topology. It investigates the differential equation aspects of quantum cohomology, before moving on to noncommutative geometry. This is followed by a further exploration of quantum field theory and gauge theory, describing AdS/CFT correspondence, and the functional renormalization group approach to quantum gravity. The second part covers a wide spectrum of topics on the borderline of mathematics and physics, ranging from orbifolds to quantum indistinguishability and involving a manifold of mathematical tools borrowed from geometry, algebra and analysis. Each chapter presents introductory material before moving on to more advanced results. The chapters are self-contained and can be read independently of the rest.

Dirichlet Branes and Mirror Symmetry

Author : Anonim
Publisher : American Mathematical Soc.
Page : 698 pages
File Size : 40,8 Mb
Release : 2009
Category : Mathematics
ISBN : 9780821838488

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Dirichlet Branes and Mirror Symmetry by Anonim Pdf

Research in string theory has generated a rich interaction with algebraic geometry, with exciting work that includes the Strominger-Yau-Zaslow conjecture. This monograph builds on lectures at the 2002 Clay School on Geometry and String Theory that sought to bridge the gap between the languages of string theory and algebraic geometry.

Topology, Geometry and Quantum Field Theory

Author : Ulrike Luise Tillmann
Publisher : Cambridge University Press
Page : 596 pages
File Size : 51,9 Mb
Release : 2004-06-28
Category : Mathematics
ISBN : 0521540496

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Topology, Geometry and Quantum Field Theory by Ulrike Luise Tillmann Pdf

The symposium held in honour of the 60th birthday of Graeme Segal brought together leading physicists and mathematicians. Its topics were centred around string theory, M-theory, and quantum gravity on the one hand, and K-theory, elliptic cohomology, quantum cohomology and string topology on the other. Geometry and quantum physics developed in parallel since the recognition of the central role of non-abelian gauge theory in elementary particle physics in the late seventies and the emerging study of super-symmetry and string theory. With its selection of survey and research articles these proceedings fulfil the dual role of reporting on developments in the field and defining directions for future research. For the first time Graeme Segal's manuscript 'The definition of Conformal Field Theory' is published, which has been greatly influential over more than ten years. An introduction by the author puts it into the present context.

Frobenius Algebras and 2-D Topological Quantum Field Theories

Author : Joachim Kock
Publisher : Cambridge University Press
Page : 260 pages
File Size : 44,9 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

Author : Thomas Kerler,Volodymyr V. Lyubashenko
Publisher : Springer
Page : 383 pages
File Size : 51,6 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.

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

Author : Hiro Lee Tanaka
Publisher : Springer Nature
Page : 84 pages
File Size : 55,6 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.

The Universal Coefficient Theorem and Quantum Field Theory

Author : Andrei-Tudor Patrascu
Publisher : Springer
Page : 270 pages
File Size : 44,9 Mb
Release : 2016-09-23
Category : Science
ISBN : 9783319461434

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The Universal Coefficient Theorem and Quantum Field Theory by Andrei-Tudor Patrascu Pdf

This thesis describes a new connection between algebraic geometry, topology, number theory and quantum field theory. It offers a pedagogical introduction to algebraic topology, allowing readers to rapidly develop basic skills, and it also presents original ideas to inspire new research in the quest for dualities. Its ambitious goal is to construct a method based on the universal coefficient theorem for identifying new dualities connecting different domains of quantum field theory. This thesis opens a new area of research in the domain of non-perturbative physics—one in which the use of different coefficient structures in (co)homology may lead to previously unknown connections between different regimes of quantum field theories. The origin of dualities is an issue in fundamental physics that continues to puzzle the research community with unexpected results like the AdS/CFT duality or the ER-EPR conjecture. This thesis analyzes these observations from a novel and original point of view, mainly based on a fundamental connection between number theory and topology. Beyond its scientific qualities, it also offers a pedagogical introduction to advanced mathematics and its connection with physics. This makes it a valuable resource for students in mathematical physics and researchers wanting to gain insights into (co)homology theories with coefficients or the way in which Grothendieck's work may be connected with physics.

Field Theories of Condensed Matter Physics

Author : Eduardo Fradkin
Publisher : Cambridge University Press
Page : 855 pages
File Size : 47,8 Mb
Release : 2013-02-28
Category : Science
ISBN : 9780521764445

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Field Theories of Condensed Matter Physics by Eduardo Fradkin Pdf

Presenting the physics of the most challenging problems in condensed matter using the conceptual framework of quantum field theory, this book is of great interest to physicists in condensed matter and high energy and string theorists, as well as mathematicians. Revised and updated, this second edition features new chapters on the renormalization group, the Luttinger liquid, gauge theory, topological fluids, topological insulators and quantum entanglement. The book begins with the basic concepts and tools, developing them gradually to bring readers to the issues currently faced at the frontiers of research, such as topological phases of matter, quantum and classical critical phenomena, quantum Hall effects and superconductors. Other topics covered include one-dimensional strongly correlated systems, quantum ordered and disordered phases, topological structures in condensed matter and in field theory and fractional statistics.

Geometry and Quantum Field Theory

Author : Daniel S. Freed,Karen K. Uhlenbeck,American Mathematical Society,Institute for Advanced Study (Princeton, N.J.)
Publisher : American Mathematical Soc.
Page : 476 pages
File Size : 41,6 Mb
Release : 1995
Category : Science
ISBN : 0821886835

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Geometry and Quantum Field Theory by Daniel S. Freed,Karen K. Uhlenbeck,American Mathematical Society,Institute for Advanced Study (Princeton, N.J.) Pdf

The first title in a new series, this book explores topics from classical and quantum mechanics and field theory. The material is presented at a level between that of a textbook and research papers making it ideal for graduate students. The book provides an entree into a field that promises to remain exciting and important for years to come.