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Lecture Notes on Chern-Simons-Witten the - Sen Hu

Lecture Notes on Chern-Simons-Witten the

By: Sen Hu


Published: 29th June 2001
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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.

Prefacep. vii
Examples of Quantizationsp. 1
Quantization of R[superscript 2]p. 1
Classical mechanicsp. 1
Symplectic methodp. 3
Holomorphic methodp. 6
Holomorphic representation of symplectic quotients and its quantizationp. 7
An example of circle actionp. 7
Moment map of symplectic actionsp. 9
Some geometric invariant theoryp. 11
Grassmaniansp. 12
Calabi-Yau/Ginzburg-Landau correspondencep. 13
Quantization of symplectic quotientsp. 14
Classical Solutions of Gauge Field Theoryp. 17
Moduli space of classical solutions of Chern-Simons actionp. 17
Symplectic reduction of gauge fields over a Riemann surfacep. 17
Chern-Simons action on a three manifoldp. 19
Maxwell equations and Yang-Mills equationsp. 22
Maxwell equationsp. 22
Yang-Mills equationsp. 23
Vector bundle, Chern classes and Chern-Weil theoryp. 25
Vector bundle and connectionp. 25
Curvature, Chern classes and Chern-Weil theoryp. 26
Quantization of Chern-Simons Actionp. 27
Introductionp. 27
Some formal discussions on quantizationp. 28
Pre-quantizationp. 31
M as a complex varietyp. 31
Quillen's determinant bundle on M and the Laplacianp. 32
Some Lie groupsp. 32
G = Rp. 32
G = S[superscript 1] = R/2[pi]Zp. 33
T*Gp. 34
Compact Lie groups, G = SU (2)p. 35
Genus onep. 35
Riemann sphere with puncturesp. 36
Higher genus Riemann surfacep. 38
Relation with WZW model and conformal field theoryp. 39
Independence of complex structuresp. 40
Borel-Weil-Bott theorem of representation of Lie groupsp. 44
Chern-Simons-Witten Theory and Three Manifold Invariantp. 47
Representation of mapping class group and three manifold invariantp. 47
Knizhik-Zamolodchikov equations and conformal blocksp. 48
Braiding and fusing matricesp. 50
Projective representation of mapping class groupp. 53
Three-dimensional manifold invariants via Heegard decompositionp. 57
Calculations by topological quantum field theoryp. 59
Atiyah's axiomsp. 59
An example: connected sump. 60
Jones polynomialsp. 60
Surgeryp. 61
Verlinde's conjecture and its proofp. 63
A brief survey on quantum group methodp. 64
Algebraic representation of knotp. 64
Hopf algebra and quantum groupsp. 67
Chern-Simons theory and quantum groupsp. 68
Renormalized Perturbation Series of Chern-Simons-Witten Theoryp. 71
Path integral and morphism of Hilbert spacesp. 71
One-dimensional quantum field theoryp. 71
Schroedinger operatorp. 72
Spectrum and determinantp. 75
Asymptotic expansion and Feynman diagramsp. 77
Asymptotic expansion of integrals, finite dimensional casep. 77
Integration on a sub-varietyp. 81
Partition function and topological invariantsp. 82
Gauge fixing and Faddeev-Popov ghostsp. 83
The leading termp. 85
Wilson line and link invariantsp. 88
A brief introduction on renormalization of Chern-Simons theoryp. 89
A regulization schemep. 90
The Feynman rulesp. 91
Topological Sigma Model and Localizationp. 95
Constructing knot invariants from open string theoryp. 95
Introductionp. 95
A topological sigma modelp. 96
Localization principlep. 97
Large N expansion of Chern-Simons gauge theoryp. 98
Equivariant cohomology and localizationp. 99
Equivariant cohomologyp. 99
Localization, finite dimensional casep. 100
Atiyah-Bott's residue formula and Duistermaat-Heckman formulap. 101
Complex case, Atiyah-Bott's residue formulap. 101
Symplectic case, Duistermaat-Heckman formulap. 102
2D Yang-Mills theory by localization principlep. 104
Cohomological Yang-Mills field theoryp. 104
Relation with physical Yang-Mills theoryp. 105
Evaluation of Yang-Mills theoryp. 107
Combinatorial approach to 2D Yang-Mills theoryp. 110
Complex Manifold Without Potential Theoryp. 113
Geometric Quantization of Chern-Simons Gauge Theoryp. 121
On Holomorphic Factorization of WZW and Coset Modelsp. 169
Bibliographyp. 193
Indexp. 197
Afterwardsp. 199
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9789810239091
ISBN-10: 9810239092
Audience: General
Format: Paperback
Language: English
Number Of Pages: 212
Published: 29th June 2001
Publisher: World Scientific Publishing Co Pte Ltd
Country of Publication: SG
Dimensions (cm): 21.69 x 15.24  x 0.79
Weight (kg): 0.31