Lecture Notes on Chern-Simons-Witten Theory.
Title:
Lecture Notes on Chern-Simons-Witten Theory.
Author:
Hu, Sen.
ISBN:
9789812386571
Personal Author:
Physical Description:
1 online resource (214 pages)
Contents:
Contents -- Preface -- Chapter 1 Examples of Quantizations -- 1.1 Quantization of R2 -- 1.1.1 Classical mechanics -- 1.1.2 Symplectic method -- 1.1.3 Holomorphic method -- 1.2 Holomorphic representation of symplectic quotients and its quantization -- 1.2.1 An example of circle action -- 1.2.2 Moment map of symplectic actions -- 1.2.3 Some geometric invariant theory -- 1.2.4 Grassmanians -- 1.2.5 Celabi-Yau/Ginzburg-Landau correspondence -- 1.2.6 Quantization of symplectic quotients -- Chapter 2 Classical Solutions of Gauge Field Theory -- 2.1 Moduli space of classical solutions of Chern-Simons action -- 2.1.1 Symplectic reduction of gauge fields over a Riemann surface -- 2.1.2 Chern-Simons action on a three manifold -- 2.2 Maxwell equations and Yang-Mills equations -- 2.2.1 Maxwell equations -- 2.2.2 Yang-Mills equations -- 2.3 Vector bundle, Chern classes and Chern-Weil theory -- 2.3.1 Vector bundle and connection -- 2.3.2 Curvature, Chern, classes and Chern- Weil theory -- Chapter 3 Quantization of Chern-Simons Action -- 3.1 Introduction -- 3.2 Some formal discussions on quantization -- 3.3 Pre-quantization -- 3.3.1 M as a complex variety -- 3.3.2 Quillen's determinant bundle o n M and the Laplacian -- 3.4 Some Lie groups -- 3.4.1 G = R -- 3.4.2 G = S1 = R/2xZ -- 3.4.3 T*G -- 3.5 Compact Lie groups, G = SU(2) -- 3.5.1 Genus one -- 3.5.2 Riemann sphere with punctures -- 3.5.3 Higher genus Riemann surface -- 3.5.4 Relation with WZW model and conformal field theory -- 3.6 Independence of complex structures -- 3.7 Borel-Weil-Bott theorem of representation of Lie groups -- Chapter 4 Chern-Simons-Witten Theory and Three Manifold Invariant -- 4.1 Representation of mapping class group and three manifold invariant -- 4.1.1 Knizhik-Zamolodchikov equations and conformal blocks -- 4.1.2 Braiding and fusing matrices.
4.1.3 Projective representation of mapping class group -- 4.1.4 Three-dimensional manifold invariants via Heegard decomposition -- 4.2 Calculations by topological quantum field theory -- 4.2.1 Atiyah's axioms -- 4.2.2 An example: connected sum -- 4.2.3 Jones polynomials -- 4.2.4 Surgery -- 4.2.5 Verlinde's conjecture and its proof -- 4.3 A brief survey on quantum group method -- 4.3.1 Algebraic representation of knot -- 4.3.2 Hopf algebra and quantum groups -- 4.3.3 Chern-Simons theory and quantum groups -- Chapter 5 Renormalized Perturbation Series of Chern-Simons- Witten Theory -- 5.1 Path integral arid morphism of Hilbert spaces -- 5.1.1 One-dimensional quantum field theory -- 5.1.2 Schroedinger operator -- 5.1.3 Spectrum and determinant -- 5.2 Asymptotic expansion and Feynman diagrams -- 5.2.1 Asymptotic expansion of integrals, finite dimensional case -- 5.2.2 Integration on a sub-variety -- 5.3 Partition function and topological invariants -- 5.3.1 Gauge fixing and Faddeev-Popov ghosts -- 5.3.2 The leading term -- 5.3.3 Wilson line and link invariants -- 5.4 A brief introduction an renormalization of Chern-Simons theory -- 5.4.1 A regulization scheme -- 5.4.2 The Feynman rules -- Chapter 6 Topological Sigma Model and Localization -- 6.1 Constructing knot invariants from open string theory -- 6.1.1 Introduction -- 6.1.2 A topological sigma model -- 6.1.3 Localization principle -- 6.1.4 Large N expansion of Chern-Simons gauge theory -- 6.2 Equivariant cohomology and localization -- 6.2.1 Equivariant cohomology -- 6.2.2 Localization, finite dimensional case -- 6.3 Atiyah-Bott's residue formula and Duistermaat-Heckman formula -- 6.3.1 Complex case, Atiyah-Bott's residue formula -- 6.3.2 Symplectic case, Duistermaat-Heckman formula -- 6.4 2D Yang-Mills theory by localization principle -- 6.4.1 Cohomological Yang-Mills field theory.
6.4.2 Relation with physical Yang-Mills theory -- 6.4.3 Evaluation of Yang-Mills theory -- 6.5 Combinatorial approach to 2D Yang-Mills theory -- Complex Manifold Without Potential Theory by S. S. Chern -- GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY by S. Axelrod, S. D. Pietra and E. Witten -- On Holomorphic Factorization of WZW and Coset Models -- Bibliography -- Index -- Afterwards.
Abstract:
This invaluable monograph has arisen in part from E Witten's lectures on topological quantum field theory in the spring of 1989 at Princeton University. At that time Witten unified several important mathematical works in terms of quantum field theory, most notably the Donaldson polynomial, the Gromov-Floer homology and the Jones polynomials. In his lectures, among other things, Witten explained his intrinsic three-dimensional construction of Jones polynomials via Chern-Simons gauge theory. He provided both a rigorous proof of the geometric quantization of the Chern-Simons action and a very illuminating view as to how the quantization arises from quantization of the space of connections. He constructed a projective flat connection for the Hilbert space bundle over the space of complex structures, which becomes the Knizhik-Zamolodchikov equations in a special case. His construction leads to many beautiful applications, such as the derivation of the skein relation and the surgery formula for knot invariant, a proof of Verlinde's formula, and the establishment of a connection with conformal field theory. In this book, Sen Hu has added material to provide some of the details left out of Witten's lectures and to update some new developments. In Chapter 4 he presents a construction of knot invariant via representation of mapping class groups based on the work of Moore-Seiberg and Kohno. In Chapter 6 he offers an approach to constructing knot invariant from string theory and topological sigma models proposed by Witten and Vafa. The localization principle is a powerful tool to build mathematical foundations for such cohomological quantum field theories. In addition, some highly relevant material by S S Chern and E Witten has been included as appendices for the convenience of readers: (1) Complex Manifold without Potential Theory by S S Chern, pp148-154.
(2) "Geometric quantization of Chern-Simons gauge theory" by S Axelrod, S D Pietra and E Witten. (3) "On holomorphic factorization of WZW and Coset models" by E Witten. Contents: Examples of Quantizations; Classical Solutions of Gauge Field Theory; Quantization of Chern-Simons Action; Chern-Simons-Witten Theory and Three Manifold Invariant; Renormalized Perturbation Series of Chern-Simons-Witten Theory; Topological Sigma Model and Localization. Readership: Senior undergraduates, postgraduates and researchers in mathematics and physics.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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