Dynamics on Lorentz Manifolds

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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.

Geometric and Topological Methods for Quantum Field Theory (Contemporary Mathematics)

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Geometric and Topological Methods for Quantum Field Theory (Contemporary Mathematics)

Manufacturer: American Mathematical Society
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Binding: Paperback

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ASIN: 0821840622

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Geometry of Nonlinear Field Theories
Roberto Percacci
Manufacturer: World Scientific Pub Co Inc
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Binding: Hardcover

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The Interface of Knots and Physics: American Mathematical Society Short Course January 2-3, 1995 San Francisco, California (Proceedings of Symposia in Applied Mathematics)

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The Interface of Knots and Physics: American Mathematical Society Short Course January 2-3, 1995 San Francisco, California (Proceedings of Symposia in Applied Mathematics)

Manufacturer: American Mathematical Society
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Binding: Hardcover

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Book Description

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.

Customer Reviews:

Not good at all.......2002-04-24

Want to learn topological field theory? Better read Witten's original paper and other papers on geometric quantization. This book is extremely unclear, not well-organized, with a number of mistakes.

Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners (Lecture Notes in Mathematics)

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Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners (Lecture Notes in Mathematics)
Thomas Kerler , and Volodymyr V. Lyubashenko
Manufacturer: Springer
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ASIN: 3540424164

Book Description

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.

Quantum Topology and Global Anomalies (Advanced Series in Mathematical Physics, Vol 23)

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A good overview

Quantum Topology and Global Anomalies (Advanced Series in Mathematical Physics, Vol 23)
Randy A. Baadhio
Manufacturer: World Scientific Pub Co Inc
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Binding: Hardcover

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ASIN: 9810227264

Book Description

Anomalies are ubiquitous features in quantum field theories. They can ruin the consistency of such theories and put significant restrictions on their viability, especially in dimensions higher than four. Global gauge and gravitational anomalies are to date, one of the scant powerful and probing tools available to physicists in the pursuit of uniqueness.

This monograph is one of the very few that specializes in the study of global anomalies in quantum field theories. A discussion of various issues associated to three dimensional physics - the Chern-Simons-Witten theories - widen the scope of this book. Topics discussed here comprises: the ongoing quest for three-manifolds invariant, the role of the mapping class groups in (a) the detection and cancellation of global anomalies, (b) formulating three-manifolds invariant; the geometric quantization of Chern-Simons-Witten theories; deformation quantization; study of chiral and gravitational anomalies; anomalies and the Atiyah-Patodi-Singer Index theorem; exotic spheres; global gravitational anomalies in some six and ten dimensional supergravity and superstring theories, with an additional case study of Witten SU(2) Global Gauge Anomalies.

In addition, five chapters lay out the mathematical basis for a thorough use of the topics above. One chapter focuses on the relationship between Teichmuller spaces, moduli spaces and mapping class groups. Another chapter is devoted to mapping class groups and arithmetic groups. Gauge theories on Riemann surfaces are studies in well over two chapters, the first one centered on the theory of bundles and the second on connections.

Many readers will find this a useful book, especially theoretical physicists and mathematicians. The material presented here will be of interest to both the experts who will find complete, detailed and precise descriptions of important topics of current interest in mathematical physics, and to students and newcomers to the field, who will appreciate the vast amount of information provided here, especially on global anomalies.

Customer Reviews:

A good overview.......2003-04-05

It is now a tautology that physics and mathematics are intertwined to a degree that goes far beyond saying merely that physics makes use of various results in mathematics. The latter has been going on ever since people have been doing physics. Now though ideas from physics have been finding applications in mathematics, and many exciting results in mathematics have appeared in the last twenty years due to this. This book is a brief overview of some of these at the time of publication, which is called 'topological quantum field theory' or 'quantum topology'.

The author introduces chapter 1 as a quest for obtaining invariants of 3-manifolds. Noting that the Euler characteristic is zero for all closed 3-manifolds, he then looks at hyperbolic 3-manifolds, and studies the volume-, Chern-Simons-, and the eta-invariants of these. The volume invariant is not fine enough as there exist non-homeomorphic hyperbolic 3-manifolds with equal volumes. The Chern-Simons invariant is the integral of the pullback of a 3-form on the (oriented) frame bundle of the manifold. The author quotes, but does not prove the Meyerhoff-Ruberman theorem, which gives criteria under which hyperbolic 3-manifolds with equal volumes have different Chern-Simons invariants. The eta-invariant involves eigenvalues of the Laplace operator and is given by the Atiyah-Patodi-Singer formula. There exists hyperbolic 3-manifolds with equal Chern-Simons invariants but different eta-invariants, but the author does not discuss examples. The author then summarizes briefly the quantum field-theoretic formulation of the Chern-Simons invariant, which, because of the problems with a rigorous foundation for functional integration, gives readers their first taste of 'physical mathematics'. He also discusses briefly the 'mutant' manifolds which are not distinguishable by any of these invariants, and conjectures that a more in-depth understanding of the 3-D mapping class group will allow finer invariants.

Chapter 2 is then an introduction to how to obtain 3-manifold invariants using the mapping class group using the holonomy of the Knizhink-Zamolodchikov monodromy equation and Moore-Seiberg conformal field theory. The 3-manifolds are required to have Heegaard decompositions, and the (Kohno) formula for the invariant involves a homomorphism from the mapping class group of a Riemann surface with a certain genus to the group of isotopy classes of orientation preserving self-diffeomorphisms.

Chapter 3 is an introduction to the Teichmuller and moduli spaces. The quotient group of Teichmuller space modulo the action of the mapping class group is moduli space. The description of Teichmuller space in terms of Fenchel-Nielsen coordinates is given, and the author shows how to describe the Deligne-Mumford compactification using these coordinates. He also shows the relation between the homologies of the moduli space and the mapping class group.

Chapter 4 is an overview of to what extent mapping class groups act like arithmetic groups. The author proves that the mapping class group cannot be arithmetic when the genus is greater than or equal to 3. He also addresses the question as the stability of the homology of mapping class group as the genus gets larger. He states, but does not prove, Harer's theorem, which says that the kth-homology group is independent of the genus g if g is greater than or equal to 3k + 1.

Chapter 5 is very interesting, in that it shows how to view Teichmuller space as a symplectic manifold, this being done via the Weil-Petersson form. He uses this to prove that the Deligne-Mumford compactification is projective.

Chapters 6 and 7 are quick reviews of gauge theories on Riemann surfaces. Chapter 8 then considers the geometric quantization of Chern-Simon-Witten theories. The presentation of this is excellent because the reader can see clearly the reason for employing the "Kahler polarization", which seems mysterious when first encountered. The U(1) (torus case) for geometric quantization is given before taking on the non-Abelian case.

The author finally gets to the connection with anomalies in chapter 9, wherein he discusses deformation quantization, mostly in relation to his own work. Global anomalies are viewed as being induced by an obstruction to patching a local deformation "quantizable *-product" to a global *-product. Physicists well-versed in quantum gauge theories will understand fully the appearance of the Jacobi identity here and its role in inducing global anomalies. The Wess-Zumino consistency conditions are discussed here also.

Things get more in touch with physics in chapter 10, wherein the author discusses the famous chiral and gravitational anomalies. The Green-Schwarz anomaly cancellation mechanism in heterotic superstring theory is discussed in detail. These results are connected to index theorems in the next chapter, where the author also introduces the reader to characteristic classes. The geometric interpretation of anomalies as being the curvature and holonomy of the connection in the determinant bundle is readily apparent. In addition, the reader can see the origin of the famous Fujikawa change of measure in the path integral for the effective action of a massless Dirac fermion in Euclidean space of even dimension.

The author goes on to give in his words an exhaustive account of global anomalies in chapter 12. Arising because of the failure of the symmetries of the full diffeomorphism group to be respected after quantization, the author differentiates between global gauge and gravitational anomalies, and the connection of the latter with the mapping class group. Chapter 13 then concentrates more on this connection, and with the connection to Chern-Simons-Witten theories. The discussion revolves around 3-dimensional handlebodies and due to this connection with topology is very interesting. The role of exotic differentiable structures in anomalies is made readily apparent.

The last chapter of the book is written by the topologist Louis Kauffman and overviews the results from the theory of characteristic classes needed for a study of exotic spheres. The existence of exotic structures is fascinating but their role (outside of anomaly detection) in physics still remains to be seen.

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Differential Topology and Quantum Field Theory
Charles Nash
Manufacturer: Academic Press
ProductGroup: Book
Binding: Paperback
ASIN: B000OHBAQM

Book Description

This is a deeply felt and highly informed essay collection about life in the American west by one of the finest writers ever to emerge from that region. As the Seattle Times has said of Owning It All: "You may never again see the American west in quite the same way if you take the time to view it through the eyes of William Kittredge. [This is a] stunning book." Having grown up on his family's cattle ranch in eastern Oregon, Kittredge directly confronts the contradictions and myths that lie at the heart of the Western experience: male freedom and female domesticity, the wild and the tame, self-interest and love of the land.

Customer Reviews:

Probably one of the best books I've read..........1999-09-06

A collection of essays that have appeared in various magazines & journals, Kittredge does a wonderful job painting a picture of Warner Valley and the American West. He makes it easy to understand how anyone could dream of traveling West in the hope of finding a new way of life. Easy to pick up but impossibe to put down!

The Best Book Ever Written About the Warner Valley!.......1999-03-17

The title says it all! If you've ever slept in the high country of Southeastern Oregon and been awakened by the brilliance of the moon, the mournful hooting of an owl or a coyote's howl at a time too late to remember but too early to get up, and while trying to get back to sleep on ground too hard and cold realized that we can never OWN the land, we only exist as part of it, you will appreciate Kittredge's eloquence in describing his own family's ultimately self-defeating attempts to do just that. This is Lake County as it was and is...a world apart from the Cool Green Vacationland of Western Oregon...where everything is connected to everything else, and Owning It All may be the only way to wrest a living from the land, but becomes an ephermeral concept that comes to no good end. History, geography, personal biography...an underappreciated book by a master whose prose is as tight as your puckered lips when it's 14 with a 45 mile an hour north wind on a late October morning in the Catlow Valley.

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