Topological Quantum Field Theory and Four Manifolds by Jose LabastidaTopological Quantum Field Theory and Four Manifolds by Jose Labastida

Topological Quantum Field Theory and Four Manifolds

byJose Labastida, Marcos Marino

Paperback | October 28, 2010

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The present book is the first of its kind in dealing with topological quantum field theories and their applications to topological aspects of four manifolds. It is not only unique for this reason but also because it contains sufficient introductory material that it can be read by mathematicians and theoretical physicists. On the one hand, it contains a chapter dealing with topological aspects of four manifolds, on the other hand it provides a full introduction to supersymmetry. The book constitutes an essential tool for researchers interested in the basics of topological quantum field theory, since these theories are introduced in detail from a general point of view. In addition, the book describes Donaldson theory and Seiberg-Witten theory, and provides all the details that have led to the connection between these theories using topological quantum field theory. It provides a full account of Witten's magic formula relating Donaldson and Seiberg-Witten invariants. Furthermore, the book presents some of the recent developments that have led to important applications in the context of the topology of four manifolds.

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Title:Topological Quantum Field Theory and Four ManifoldsFormat:PaperbackDimensions:234 pages, 9.45 × 6.3 × 0.68 inPublished:October 28, 2010Publisher:Springer NetherlandsLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9048167795

ISBN - 13:9789048167791

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Table of Contents

Table of Contents Preface vii1. Topological Aspects of Four-Manifolds 11.1. Homology and cohomology 11.2. The intersection form 21.3. Self-dual and anti-self-dual forms 41.4. Characteristic classes 51.5. Examples of four-manifolds. Complex surfaces 61.6. Spin and Spinc-structures on four-manifolds. 92. The Theory of Donaldson Invariants 122.1. Yang-Mills theory on a four-manifold 122.2. SU(2) and SO(3) bundles 142.3. ASD connections 162.4. Reducible connections 182.5. A local model for the moduli space 192.6. Donaldson invariants 222.7. Metric dependence 273. The Theory of Seiberg-Witten Invariants 313.1. The Seiberg-Witten equations 313.2. The Seiberg-Witten invariants 323.3. Metric dependence 364. Supersymmetry in Four Dimensions 394.1. The supersymmetry algebra 394.2. N = 1 superspace and super.elds 404.3. N = 1 supersymmetric Yang-Mills theories 454.4. N = 2 supersymmetric Yang-Mills theories 504.5. N = 2 supersymmetric hypermultiplets 534.6. N = 2 supersymmetric Yang-Mills theories with matter 555. Topological Quantum Field Theories in Four Dimensions 585.1. Basic properties of topological quantum .eld theories 585.2. Twist of N = 2 supersymmetry 615.3. Donaldson-Witten theory 645.4. Twisted N = 2 supersymmetric hypermultiplet 715.5. Extensions of Donaldson-Witten theory 725.6. Monopole equations 746. The Mathai-Quillen Formalism 786.1. Equivariant cohomology 796.2. The .nite-dimensional case 826.3. A detailed example 886.4. Mathai-Quillen formalism: In.nite-dimensional case 936.5. The Mathai-Quillen formalism for theories with gauge symmetry 1026.6. Donaldson-Witten theory in the Mathai-Quillen formalism 1056.7. Abelian monopoles in the Mathai-Quillen formalism 1077. The Seiberg-Witten Solution of N = 2 SUSY Yang-Mills Theory 1107.1. Low energy e.ective action: semi-classical aspects 1107.2. Sl(2,Z) duality of the e.ective action 1167.3. Elliptic curves 1207.4. The exact solution of Seiberg and Witten 1237.5. The Seiberg-Witten solution in terms of modular forms 1298. The u-plane Integral 1338.1. The basic principle (or, 'Coulomb + Higgs=Donaldson') 1338.2. E.ective topological quantum .eld theory on the u-plane 1348.3. Zero modes 1408.4. Final form for the u-plane integral 1448.5. Behavior under monodromy and duality 1499. Some Applications of the u-plane Integral 1549.1. Wall crossing 1549.2. The Seiberg-Witten contribution 1579.3. The blow-up formula. 16510. Further Developments in Donaldson-Witten Theory 17010.1. More formulae for Donaldson invariants 17010.2. Applications to the geography of four-manifolds 17710.3. Extensions to higher rank gauge groups 188Appendix A. Spinors in Four Dimensions 204Appendix B. Elliptic Functions and Modular Forms 209Bibliography 213

Editorial Reviews

From the reviews of the first edition:"The present book . starts with a survey of important topological topics, then reviews the theories of Donaldson and Seiberg-Witten, and describes various aspects of supersymmetery . . Graduate students, post-docs and junior faculty interested in the interaction of physics and mathematics . will greatly benefit from this coherent treatment of the subject and the thorough evaluation of its virtues which is, to my knowledge, the first of its kind." (Gert Roepstorff, Zentralblatt MATH, Vol. 1087, 2006)"The book is written to be accessible either for physicists who want to know about the topological consequences of supersymmetric quantum field theory, or for mathematicians curious about where the links between Donaldson and Seiberg-Witten theory come from. For both groups it should be a good point of entry to the literature. . the authors manage to give an end-to-end treatment of the relation between Donaldson and Seiberg-Witten invariants, including a detailed computation of the formula connecting the two for manifolds of Seiberg-Witten simple type." (Andrew Neitzke, Mathematical Reviews, Issue 2006 f)