Conformal Field Theory And Topology

Author : Toshitake Kohno
Publisher : American Mathematical Soc.
Page : 188 pages
File Size : 55,9 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.

Author : Toshitake Kohno
Publisher : Unknown
Page : 128 pages
File Size : 52,7 Mb
Release : 2002
Category : Conformal invariants
ISBN : 1470446359

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Conformal Field Theory and Topology by Toshitake Kohno Pdf

The aim of this book is to provide the reader with an introduction to conformal field theory and its applications to topology. The author starts with a description of geometric aspects of conformal field theory based on loop groups. By means of the holonomy of conformal field theory he defines topological invariants for knots and 3-manifolds. He also gives a brief treatment of Chern-Simons perturbation theory.

Author : Ulrike Luise Tillmann
Publisher : Cambridge University Press
Page : 596 pages
File Size : 47,5 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.

Author : Daniel S. Freed
Publisher : American Mathematical Soc.
Page : 186 pages
File Size : 44,6 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.

Author : Charles Nash
Publisher : Elsevier
Page : 404 pages
File Size : 45,5 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

Author : Martin Schottenloher
Publisher : Springer Science & Business Media
Page : 153 pages
File Size : 42,7 Mb
Release : 2008-09-15
Category : Science
ISBN : 9783540706908

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A Mathematical Introduction to Conformal Field Theory by Martin Schottenloher Pdf

Part I gives a detailed, self-contained and mathematically rigorous exposition of classical conformal symmetry in n dimensions and its quantization in two dimensions. The conformal groups are determined and the appearence of the Virasoro algebra in the context of the quantization of two-dimensional conformal symmetry is explained via the classification of central extensions of Lie algebras and groups. Part II surveys more advanced topics of conformal field theory such as the representation theory of the Virasoro algebra, conformal symmetry within string theory, an axiomatic approach to Euclidean conformally covariant quantum field theory and a mathematical interpretation of the Verlinde formula in the context of moduli spaces of holomorphic vector bundles on a Riemann surface.

Author : Michio Kaku
Publisher : Springer Science & Business Media
Page : 542 pages
File Size : 47,7 Mb
Release : 2012-12-06
Category : Science
ISBN : 9781461205036

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Strings, Conformal Fields, and M-Theory by Michio Kaku Pdf

Building on the foundations laid in his Introduction to Superstrings and M Theory, Professor Kaku discusses such topics as the classification of conformal string theories, knot theory, the Yang-Baxter relation, quantum groups, and the insights into 11-dimensional strings recently obtained from M-theory. New chapters discuss such topics as Seiberg-Witten theory, M theory and duality, and D-branes. Throughout, the author conveys the vitality of the current research and places readers at its forefront. Several chapters reviewing the fundamentals of string theory, making the presentation of the material self-contained while keeping overlap with the earlier book to a minimum.

Author : Kenji Ueno
Publisher : American Mathematical Soc.
Page : 178 pages
File Size : 44,7 Mb
Release : 2008
Category : Conformal invariants
ISBN : 9780821840887

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Conformal Field Theory with Gauge Symmetry by Kenji Ueno Pdf

This book presents a systematic approach to conformal field theory with gauge symmetry from the point of view of complex algebraic geometry. After presenting the basic facts of the theory of compact Riemann surfaces and the representation theory of affine Lie algebras in Chapters 1 and 2, conformal blocks for pointed Riemann surfaces with coordinates are constructed in Chapter 3. In Chapter 4 the sheaf of conformal blocks associated to a family of pointed Riemann surfaces withcoordinates is constructed, and in Chapter 5 it is shown that this sheaf supports a projective flat connection-one of the most important facts of conformal field theory. Chapter 6 is devoted to the study of the detailed structure of the conformal field theory over $\mathbb{P}1$.Recently it was shown that modular functors can be constructed from conformal field theory, giving an interesting relationship between algebraic geometry and topological quantum field theory. This book provides a timely introduction to an intensively studied topic of conformal field theory with gauge symmetry by a leading algebraic geometer, and includes all the necessary techniques and results that are used to construct the modular functor.

Author : Michio Kaku
Publisher : Unknown
Page : 560 pages
File Size : 40,6 Mb
Release : 1991
Category : Mathematics
ISBN : UOM:39015021876977

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Strings, Conformal Fields, and Topology by Michio Kaku Pdf

Author : Vladimir G. Turaev
Publisher : European Mathematical Society
Page : 300 pages
File Size : 55,9 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.

Author : Hendrik B. Geyer
Publisher : Springer
Page : 232 pages
File Size : 43,9 Mb
Release : 1995-08-18
Category : Mathematics
ISBN : UOM:39015034924038

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Field Theory, Topology and Condensed Matter Physics by Hendrik B. Geyer Pdf

This topical volume contains five pedagogically written articles on the interplay between field theory and condensed matter physics. The main emphasis is on the topological aspects, and especially quantum Hall fluids, and superconductivity is treated extensively. Other topics are conformal invariance and path integrals. The articles are carefully edited so that the book could ideally serve as a text for special graduate courses.

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 : 44,6 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.

Author : Ralph Blumenhagen,Erik Plauschinn
Publisher : Springer
Page : 265 pages
File Size : 50,8 Mb
Release : 2009-07-11
Category : Science
ISBN : 9783642004506

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Introduction to Conformal Field Theory by Ralph Blumenhagen,Erik Plauschinn Pdf

Based on class-tested notes, this text offers an introduction to Conformal Field Theory with a special emphasis on computational techniques of relevance for String Theory. It introduces Conformal Field Theory at a basic level, Kac-Moody algebras, one-loop partition functions, Superconformal Field Theories, Gepner Models and Boundary Conformal Field Theory. Eventually, the concept of orientifold constructions is explained in detail for the example of the bosonic string. In providing many detailed CFT calculations, this book is ideal for students and scientists intending to become acquainted with CFT techniques relevant for string theory but also for students and non-specialists from related fields.

Author : Sergei V Ketov
Publisher : World Scientific
Page : 502 pages
File Size : 48,8 Mb
Release : 1995-02-28
Category : Science
ISBN : 9789814502528

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Conformal Field Theory by Sergei V Ketov Pdf

Conformal field theory is an elegant and powerful theory in the field of high energy physics and statistics. In fact, it can be said to be one of the greatest achievements in the development of this field. Presented in two dimensions, this book is designed for students who already have a basic knowledge of quantum mechanics, field theory and general relativity. The main idea used throughout the book is that conformal symmetry causes both classical and quantum integrability. Instead of concentrating on the numerous applications of the theory, the author puts forward a discussion of the general methods of conformal field theory as a physical theory. Hence the book provides in a self-contained way the necessary knowledge and “conformal” intuition which underline the various applications of conformal field theory. It is aimed to assist students and professionals in the study of the theory from its first principles and in applying the methods in their own research. The first of its kind, this book promises to give a detailed and comprehensive insight into the workings of conformal field theory.

Author : Bojko Bakalov,Alexander A. Kirillov
Publisher : American Mathematical Soc.
Page : 232 pages
File Size : 46,8 Mb
Release : 2001
Category : Abelian categories
ISBN : 9780821826867

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Lectures on Tensor Categories and Modular Functors by Bojko Bakalov,Alexander A. Kirillov Pdf

This book gives an exposition of the relations among the following three topics: monoidal tensor categories (such as a category of representations of a quantum group), 3-dimensional topological quantum field theory, and 2-dimensional modular functors (which naturally arise in 2-dimensional conformal field theory). The following examples are discussed in detail: the category of representations of a quantum group at a root of unity and the Wess-Zumino-Witten modular functor. The idea that these topics are related first appeared in the physics literature in the study of quantum field theory. Pioneering works of Witten and Moore-Seiberg triggered an avalanche of papers, both physical and mathematical, exploring various aspects of these relations. Upon preparing to lecture on the topic at MIT, however, the authors discovered that the existing literature was difficult and that there were gaps to fill. The text is wholly expository and finely succinct. It gathers results, fills existing gaps, and simplifies some proofs. The book makes an important addition to the existing literature on the topic. It would be suitable as a course text at the advanced-graduate level.