Topology, Geometry, and Gauge Fields: Interactions (Applied Mathematical Sciences)

Book Description

This book covers topology and geometry beginning with an accessible account of the extraordinary and rather mysterious impact of mathematical physics, especially gauge theory, on the study of the geometry and topology of manifolds. Much of the mathematics developed in the book to study the classical field theories of physics (de Rham cohomology, Chern classes, Semi-Riemannian manifolds, Cech cohomology, spinors etc.) is standard, but the treatment always keeps one eye on the physics and unhesitatingly sacrifices generality to clarity. The author brings the reader up to the level needed to conclude with a brief discussion of the Seiberg-Witten invariants. Although this volume can be read independently Naber carries on the program initiated in his earlier volume, Topology, Geometry and Gauge Fields: Foundations, Springer, 1997, and writes in much the same spirit with precisely the same philosophical motivation. A large number of exercises are included to encourage active participation on the part of the reader. This work will be of great interest to researchers and graduate students in the field of mathematical physics. REVIEWS OF TOPOLOGY, GEOMETRY, AND GAUGE FIELDS: FOUNDATIONS "It is unusual to find a book so carefully tailored to the needs of this interdisciplinary area of mathematical physics...Naber combines a knowledge of his subject with an excellent informal writing style." NZMS NEWSLETTER "...this book should be very interesting for mathematicians and

Customer Reviews:

correction to dost.......2006-05-20

The review "Easy reading, complete proofs, plenty of exercises, October 29, 2005 by Rehan Dost is of the first volume, Foundations, not this volume which is Interactions. Naber's books are crafted to bridge physics, undergraduate mathematics and graduate mathematics. This is one more of his beautiful volumes in applied mathematics.

Easy reading, complete proofs, plenty of exercises.......2005-10-30

This text is by far the best introductory text marrying basic concepts of physics with pure mathematics.

Some background in the basic concepts of vector calculus, linear algebra, complex numbers and group theory is required.

The author begins by motivating the mathematics by the pursuit of finding a vector potential to represent a magnetic monopole. We see that the topology of R3-0 precludes such a vector potential from existing. We see here a simple example of how the topology of a space affects the physics associated with it.
The importance of the vector potential as something other than a convenient computational tool is highlighted by a reference to essential inclusion in quantum mechanics. Thus we NEED such a potential.

The author now asks whether there is a "trick" or device to get around this difficulty. The device are principal bundles and connections. For example the potentials noted above must keep track of the phase of a charged test particle as it moves thru the field of a magnetic monopole. We need a "bundle" of circles ( representing the phase at each point ) over S2 ( the author explains why we need only consider S2 instead of R3-0, briefly we need only keep track of 2 of the 3 spherical co-ordinates ).
Thus a curve in S2 thought of as the particles trajectory will have to be "lifted" to the bundle space by a lifting procedure called a connection.
In a more general setting elementary particles have an internal structure ( spin etc ) which becomes apparent during interactions although may not be apparent in uniform motion thru a vacuum. Since the phase of the particle does not alter the modulus when calculating probabilities these do not change. However, when the particles interact phase differences are important. We need to keep track of such phases as the particles interact.

Thus we need a "bundle" over a 4-manifold ( keeps track of the particles space-time path ) to keep track of such internal states. One sees we also need a group to transform states into one another ( usually incorporated into the bundle ). Connections then model physical phenomena which mediate changes in the internal states.
We see that some connections satisfy the Yang-Mills equations and using the appropriate equivalence relation form Moduli spaces.

Now that may seem like alot to digest with only a spattering of mathematical maturity.

The beauty of the book is that the author starts from FIRST principles.

Chapter 1 introduces topological concepts of topology, continuity, quotient topology, projective spaces, compactness, connectivity, covering spaces and topological groups.

Chapter 2 introduces concepts of path lifting, fundamental groups, contractability, simple connectedness, covering homotopy theorem, higher homotopy groups

Chapter 3 introduces principle bundles, transition functions, bundle maps and principle bundles over spheres.

Chapter 4 introduces manifolds, derivatives on manifolds, tangent/cotangent spaces, submanifolds, vector fields, matrix lie groups, vector valued 1- forms, 2 forms and Riemann metrics

Chapter 5 gets to some physics with gauge fields and connections, curvature, Yang-Mills functional, moduli spaces, Hodge dual , matter fields and covariant derivatives.

At each step the author carefully provides complete proofs and easy exercises to ensure understanding.

It was a pleasure to read the book and complete the exercises. At no point did I feel frustration or boredom.

Don't waste your money.......2004-08-27

This review refers only to the book printing quality not to the contents.

I had purchased some books from Springer in the past (Like Arnold Mathematical Methods of Classical Mechanics, Lang Algebra etc..) and found them beautifully edited: good binding, paper etc..

And to my surprise I was very disappointed with the overall quality of this book, poor binding -glued instead of sewn- bad quality paper -forming waves at the binding spine, etc..

You pay for a quality item, a book you can use for years, and you get a hardbound crap that you can not left open in a table without holding it tight risking to lose the pages after a few days of use in the process.

I find this unacceptable in books costing 60$+. Sadly I find this to occur very often, publishers should be more careful with their printings and custumers should demand a better quality.

Don't waste your money.

A reader.

MATH AND TOPOLOGY.......2001-05-08

Topology is very important scince in the fields of mathematics. And it using in many of another sinceis.

required reading for a topologist interested in physics.......2000-05-14

As a mathematician turned physics grad student, it is often difficult to read "Math for Physicists" books simply because of the focus on making "numbers churn out;" which, at least for me personally, more difficult to get a handle on the subject and then, in turn, use it fruitfully.

This book on the other hand, is exemplary of why I got into physics in the first place. The first chapter (Physical motivations) and the last chapter (Gauge Fields and Instantons) can be read by any one with undergraduate topology under their belt and come away with a more powerful understanding of gauge theory than, in my opinion, can be found in other introductory gauge theory texts I've been directed to.

Book Description

This monograph deals with the geometrical and topological aspects associated with the quantization procedure, and it is shown how these features are manifested in anomaly and Berry Phase. This book is unique in its emphasis on the topological aspects of a fermion which arise as a consequence of the quantization procedure. Also, an overview of quantization procedures is presented, tracing the equivalence of these methods by noting that the gauge field plays a significant role in all these procedures, as it contains the ingredients of topological features.
Audience: This book will be of value to research workers and specialists in mathematical physics, quantum mechanics, quantum field theory, particle physics and differential geometry.

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Conformal Field Theory and Topology
Toshitaki Kohno
Manufacturer: American Mathematical Society
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Binding: Mass Market Paperback

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ASIN: 082182130X

Book Description

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 invariant for 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 geometry as well. This book gives an accessible treatment for a rigorous construction of topological invariants originally defined as partition functions of fields on manifolds.

Book Description

This book provides an extensive and self-contained presentation of quantum and related invariants of knots and 3-manifolds. Polynomial invariants of knots, such as the Jones and Alexander polynomials, are constructed as quantum invariants, i.e. invariants derived from representations of quantum groups and from the monodromy of solutions to the Knizhnik-Zamolodchikov equation. With the introduction of the Kontsevich invariant and the theory of Vassiliev invariants, the quantum invariants become well-organized. Quantum and perturbative invariants, the LMO invariant, and finite type invariants of 3-manifolds are discussed. The Chern-Simons field theory and the Wess-Zumino-Witten model are described as the physical background of the invariants.

Frobenius Algebras and 2-D Topological Quantum Field Theories (London Mathematical Society Student Texts)

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Frobenius Algebras and 2-D Topological Quantum Field Theories (London Mathematical Society Student Texts)
Joachim Kock
Manufacturer: Cambridge University Press
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Categories for the Working Mathematician (Graduate Texts in Mathematics)

ASIN: 0521540313

Book Description

Describing a striking connection between topology and algebra, rather than only proving the theorem, this study demonstrates how the result fits into a more general pattern. Throughout the text 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. Includes numerous exercises and examples.

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This 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.

Quantum Field Theory and Topology (Grundlehren der mathematischen Wissenschaften)

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Quantum Field Theory and Topology (Grundlehren der mathematischen Wissenschaften)
Albert S. Schwarz
Manufacturer: Springer
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Binding: Hardcover

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ASIN: 3540547533

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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.

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Dynamics on Lorentz Manifolds
Scot Adams
Manufacturer: World Scientific Publishing Company
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Floer Homology Groups in Yang-Mills Theory (Cambridge Tracts in Mathematics)

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Floer Homology Groups in Yang-Mills Theory (Cambridge Tracts in Mathematics)
S. K. Donaldson
Manufacturer: Cambridge University Press
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Rational Homotopy Theory (Graduate Texts in Mathematics)

ASIN: 0521808030

Book Description

This monograph gives a thorough exposition of Floer's seminal work during the 1980s from a contemporary viewpoint. The material contained here was developed with specific applications in mind. However, it has now become clear that the techniques used are important for many current areas of research. An important example would be symplectic theory and gluing problems for self-dual metrics and other metrics with special holonomy. The author writes with the big picture constantly in mind. As well as a review of the current state of knowledge, there are sections on the likely direction of future research. Included in this are connections between Floer groups and the celebrated Seiberg-Witten invariants. The results described in this volume form part of the area known as Donaldson theory. The significance of this work is such that the author was awarded the prestigious Fields Medal for his contribution.

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The concept of Floer homology has been one of the most striking developments in differential geometry over the past 20 years. It yields rigorously defined invariants which can be viewed as homology groups of infinite-dimensional cycles. The ideas have led to great advances in the areas of low-dimensional topology and symplectic geometry and are intimately related to developments in Quantum Field Theory. The first half of this book gives a thorough account of Floer's construction in the context of gauge theory over 3 and 4-dimensional manifolds. The second half works out some further technical developments of the theory, and the final chapter outlines some research developments for the future - including a discussion of the appearance of modular forms in the theory. The scope of the material in this book means that it will appeal to graduate students as well as those on the frontiers of the subject.

Differential Topology and Quantum Field Theory

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Fair treatment
A mathematician with a physics background

Differential Topology and Quantum Field Theory
Charles Nash
Manufacturer: Academic Press
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Geometry, Topology and Physics, Second Edition (Graduate Student Series in Physics)

ASIN: 0125140762

Book Description

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.

Key Features
* 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

Customer Reviews:

Fair treatment.......2001-05-06

This book is written for the theoretical physicist in mind. It is somewhat out-of-date, as there have been many developments in differential topology, such as the Seiberg-Witten theory, since this book was published. However, it might still serve the reader with an introduction to these latter developments. Being just a summary there are no proofs given, and this might annoy the mathematician, but it still could be read profitably by a mathematician if they need a quick introduction to the interplay between physics and differential topology. Indeed, some of the results outlined in the book have been dubbed "physical mathematics" by mathematicians because of the lack of rigour involved in some of the constructions. Quantum field theory is not a subject at all to look for mathematical rigour. Several attempts have been made to put it on a sound mathematical foundation, by these attempts all result in a weakening of its physical predictive power.

The author introduces some basic topological notions in the first chapter, such as homotopy and homology/cohomology groups. He does give a good explanation via the smash product, of how to get a base point in a product space when each factor has a base point. Also, his discussion of H-and coH-spaces is very intuitive and serves the physicist-reader well in developing a functorial mind set. Freedman's and Donaldson's work in 4-dimensional topology is discussed only very briefly however. The existence of exotic structures on 4-dimensional topology is discussed only very briefly however. The existence of exotic structures on 4-dimensional Euclidean space has recently been shown to have interesting physical consequences, but the author only devotes a few sentences to exotica. The explicit construction of an exotic structure is of great importance to physical applications, but as of yet only existence results are known.

Physicists are used to dealing with elliptic partial differential equations, and the next chapter discusses these in a more abstract guise: the theory of elliptic operators. These are introduced in the context of vector bundles, as preparation for the Atiyah-Singer index theorem. The locality of pseudo-differential operators sets up the need for Sobolev spaces, and the author does a fairly good job of overviewing the main results.

The concept of a sheaf is introduced in the next chapter, but I think physicists would understand sheaves better if they were introduced via analytic continuation, a procedure that physicists are very well acquainted with. K-theory is also discussed in this chapter and the corresponding stable theory. Physicists have to understand the Bott periodicity theorems when doing functional integration in quantum field theory. Characteristic classes are only briefly treated, and, like all the treatments of this subject, the discussion gives no insight as to why these objects work as well as they do.

The author returns to elliptic operators in the next chapter, where their index theory is discussed. The treatment is too formal. and the reader will have to search the literature for more in-depth discussion.

Algebraic geometry, which has taken on immense importance in string theories and M-theory, is introduced in the next chapter. This chapter might be too quick for the physicist needing an understanding for applicaations in these areas. More concrete examples of varieties and explicit calculations of moduli spaces would have been helpful.

Physicists who have done current algebra will appreciate the next chapter on infinite dimensional groups. The loop group, gauge group, Virasoro group, and the Kac-Moody algebra, of use in conformal field theories and gauge field theories, are given fairly good treatment.

Morse theory, so indispensable in both mechanics and quantum field theory, is discussed in the next chapter. This is probably the best written of the chapters in the book, especially the sections on equivariance and supersymmetry.

Instantons, so important in guage theories and the subsequent quantization via functional integration, are treated in Chapter 8. It is a fairly good discussion, with infinite dimensional critical point theory given emphasis.

Applications to string theory is the subject of the next chapter, but the chapter is far too short to be of much use to someone first entering the field.

The treatment of anomalies in the next chapter is quite good though; the section on Fock space and Gauss's law is one of the best I have seen in the literature. The author explains carefully the origin of the Schwinger term.

Conformal field theories follow in Chapter 9, and the Virasoro algebra again makes its appearance. This is an area that employs more of the "hard" analysis in obtaining results rather than "soft" techniques, so physicists should be fairly comfortable with the discussion.

The last chapter introduces a topic that could fairly be classified as a "quantization of mathematics". The author discusses topological quantum field theories, and it is in this area that I believe the most fascinating constructions in all of mathematical physics take place. These theories have spurred a tremendous amount of research, and the author gives a fairly good overview. The book is a little too overpriced considering the content and the fact that it is a paperback. Such expense is worth it for a self-contained book, but this is not one of these, and must be supplemented by a great deal of outside reading.

A mathematician with a physics background.......2000-04-04

This book has a huge amount of mathematics packed inside it. It covers most of the math needed for understanding string theory, but because of its scope very few of the theorems are proved. I have found it very helpful as an overview.

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Invariants of Homology 3-Spheres
Nikolai Saveliev
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 3540437967

Book Description

Homology 3-sphere is a closed 3-dimensional manifold whose homology equals that of the 3-sphere. These objects may look rather special but they have played an outstanding role in geometric topology for the past fifty years. The book gives a systematic exposition of diverse ideas and methods in the area, from algebraic topology of manifolds to invariants arising from quantum field theories. The main topics covered are constructions and classification of homology 3-spheres, Rokhlin invariant, Casson invariant and its extensions, including invariants of Walker and Lescop, Herald and Lin invariants of knots, and equivariant Casson invariants, Floer homology and gauge-theoretical invariants of homology cobordism. Many of the topics covered in the book appear in monograph form for the first time. The book gives a rather broad overview of ideas and methods and provides a comprehensive bibliography. It will be appealing to both graduate students and researchers in mathematics and theoretical physics.

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