Topological Quantum Field Theory And Four Manifolds

Author : Jose Labastida,Marcos Marino
Publisher : Springer Science & Business Media
Page : 235 pages
File Size : 50,7 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.

Author : Hernan Ocampo,Sylvie Paycha,Andrés Vargas
Publisher : Springer Science & Business Media
Page : 256 pages
File Size : 47,5 Mb
Release : 2005-06-13
Category : Science
ISBN : 354024283X

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Geometric and Topological Methods for Quantum Field Theory by Hernan Ocampo,Sylvie Paycha,Andrés Vargas Pdf

This volume offers an introduction, in the form of four extensive lectures, to some recent developments in several active topics at the interface between geometry, topology and quantum field theory. The first lecture is by Christine Lescop on knot invariants and configuration spaces, in which a universal finite-type invariant for knots is constructed as a series of integrals over configuration spaces. This is followed by the contribution of Raimar Wulkenhaar on Euclidean quantum field theory from a statistical point of view. The author also discusses possible renormalization techniques on noncommutative spaces. The third lecture is by Anamaria Font and Stefan Theisen on string compactification with unbroken supersymmetry. The authors show that this requirement leads to internal spaces of special holonomy and describe Calabi-Yau manifolds in detail. The last lecture, by Thierry Fack, is devoted to a K-theory proof of the Atiyah-Singer index theorem and discusses some applications of K-theory to noncommutative geometry. These lectures notes, which are aimed in particular at graduate students in physics and mathematics, start with introductory material before presenting more advanced results. Each chapter is self-contained and can be read independently.

Author : Thomas Kerler,Volodymyr V. Lyubashenko
Publisher : Springer
Page : 383 pages
File Size : 55,7 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.

Author : Hernan Ocampo,Sylvie Paycha,Alexander Cardona
Publisher : World Scientific
Page : 495 pages
File Size : 46,8 Mb
Release : 2003
Category : Science
ISBN : 9789812381316

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Proceedings of the Summer School Geometric and Topological Methods for Quantum Field Theory by Hernan Ocampo,Sylvie Paycha,Alexander Cardona 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.

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 : 42,8 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.

Author : Vijay Kodiyalam
Publisher : CRC Press
Page : 138 pages
File Size : 51,8 Mb
Release : 2019-05-20
Category : Mathematics
ISBN : 142003555X

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Topological Quantum Field Theories from Subfactors by Vijay Kodiyalam Pdf

Pure mathematicians have only recently begun a rigorous study of topological quantum field theories (TQFTs). Ocneanu, in particular, showed that subfactors yield TQFTs that complement the Turaev-Viro construction. Until now, however, it has been difficult to find an account of this work that is both detailed and accessible. Topological Quant

Author : Albert S. Schwarz
Publisher : Springer Science & Business Media
Page : 277 pages
File Size : 55,5 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.

Author : Thomas Kerler,Volodymyr V. Lyubashenko
Publisher : Unknown
Page : 392 pages
File Size : 42,9 Mb
Release : 2014-09-01
Category : Electronic
ISBN : 366219435X

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

Author : John M. Bryden
Publisher : Springer Science & Business Media
Page : 353 pages
File Size : 55,9 Mb
Release : 2005-03-02
Category : Mathematics
ISBN : 9781402027703

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Advances in Topological Quantum Field Theory by John M. Bryden Pdf

Author : Louis H. Kauffman,Randy A. Baadhio
Publisher : World Scientific
Page : 400 pages
File Size : 47,5 Mb
Release : 1993
Category : Mathematics
ISBN : 981022575X

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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.

Author : Alexander Cardona,Sylvie Paycha,Hernan Ocampo
Publisher : World Scientific
Page : 492 pages
File Size : 47,8 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:

Author : Louis H. Kauffman
Publisher : American Mathematical Soc.
Page : 226 pages
File Size : 46,7 Mb
Release : 1995-12-01
Category : Knot theory
ISBN : 0821867563

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The interface of knots and physics by Louis H. Kauffman Pdf

This book is the result of an AMS Short Course on Knots and Physics that was held in San Francisco (January 1994). The range of the course went beyond knots to the study of invariants of low dimensional manifolds and extensions of this work to four manifolds and to higher dimensions. The authors use ideas and methods of mathematical physics to extract topological information about knots and manifolds. Features: A basic introduction to knot polynomials in relation to statistical link invariants. Concise introductions to topological quantum field theories and to the role of knot theory in quantum gravity. Knots and Physics would be an excellent supplement to a course on algebraic topology or a physics course on field theory.

Author : Daniel S. Freed
Publisher : American Mathematical Soc.
Page : 186 pages
File Size : 48,7 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.

Author : Alexander Cardona,Hernan Ocampo,Sylvie Paycha
Publisher : World Scientific
Page : 500 pages
File Size : 41,6 Mb
Release : 2003
Category : Mathematics
ISBN : 9812705066

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Proceedings of the Summer School Geometric and Topological Methods for Quantum Field Theory by Alexander Cardona,Hernan Ocampo,Sylvie Paycha 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.

Author : John W. Morgan
Publisher : Princeton University Press
Page : 138 pages
File Size : 52,9 Mb
Release : 2014-09-08
Category : Mathematics
ISBN : 9781400865161

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The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds. (MN-44), Volume 44 by John W. Morgan Pdf

The recent introduction of the Seiberg-Witten invariants of smooth four-manifolds has revolutionized the study of those manifolds. The invariants are gauge-theoretic in nature and are close cousins of the much-studied SU(2)-invariants defined over fifteen years ago by Donaldson. On a practical level, the new invariants have proved to be more powerful and have led to a vast generalization of earlier results. This book is an introduction to the Seiberg-Witten invariants. The work begins with a review of the classical material on Spin c structures and their associated Dirac operators. Next comes a discussion of the Seiberg-Witten equations, which is set in the context of nonlinear elliptic operators on an appropriate infinite dimensional space of configurations. It is demonstrated that the space of solutions to these equations, called the Seiberg-Witten moduli space, is finite dimensional, and its dimension is then computed. In contrast to the SU(2)-case, the Seiberg-Witten moduli spaces are shown to be compact. The Seiberg-Witten invariant is then essentially the homology class in the space of configurations represented by the Seiberg-Witten moduli space. The last chapter gives a flavor for the applications of these new invariants by computing the invariants for most Kahler surfaces and then deriving some basic toological consequences for these surfaces.