Lecture Notes On Chern-simons-witten Theory

By: Sen Hu

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Hardcover | 3 July 2001

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This monograph has arisen in part from E. Witten's lectures on topological quantum field theory given 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 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 Four he presents a construction of knot invariant via representation of mapping class groups based on the work of Moore-Seiberg and Kohno. In Chapter Six he offers an approach to constructing knot invariant from string theory and topological sigma models proposed by Witten and Vafa. In addition, relevant material by S.S. Chern and E. Witten has been included as appendices for the convenience of readers.

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Non-Fiction Science Physics Quantum Physics & Quantum Mechanics & Quantum Field Theory Classical Mathematics Wave Mechanics including Vibration & Acoustics Mathematics Geometry Differential & Riemannian Geometry Topology Booktopia Publisher Services World Scientific Publishing

Preface	p. vii
Examples of Quantizations	p. 1
Quantization of R[superscript 2]	p. 1
Classical mechanics	p. 1
Symplectic method	p. 3
Holomorphic method	p. 6
Holomorphic representation of symplectic quotients and its quantization	p. 7
An example of circle action	p. 7
Moment map of symplectic actions	p. 9
Some geometric invariant theory	p. 11
Grassmanians	p. 12
Calabi-Yau/Ginzburg-Landau correspondence	p. 13
Quantization of symplectic quotients	p. 14
Classical Solutions of Gauge Field Theory	p. 17
Moduli space of classical solutions of Chern-Simons action	p. 17
Symplectic reduction of gauge fields over a Riemann surface	p. 17
Chern-Simons action on a three manifold	p. 19
Maxwell equations and Yang-Mills equations	p. 22
Maxwell equations	p. 22
Yang-Mills equations	p. 23
Vector bundle, Chern classes and Chern-Weil theory	p. 25
Vector bundle and connection	p. 25
Curvature, Chern classes and Chern-Weil theory	p. 26
Quantization of Chern-Simons Action	p. 27
Introduction	p. 27
Some formal discussions on quantization	p. 28
Pre-quantization	p. 31
M as a complex variety	p. 31
Quillen's determinant bundle on M and the Laplacian	p. 32
Some Lie groups	p. 32
G = R	p. 32
G = S[superscript 1] = R/2[pi]Z	p. 33
T*G	p. 34
Compact Lie groups, G = SU (2)	p. 35
Genus one	p. 35
Riemann sphere with punctures	p. 36
Higher genus Riemann surface	p. 38
Relation with WZW model and conformal field theory	p. 39
Independence of complex structures	p. 40
Borel-Weil-Bott theorem of representation of Lie groups	p. 44
Chern-Simons-Witten Theory and Three Manifold Invariant	p. 47
Representation of mapping class group and three manifold invariant	p. 47
Knizhik-Zamolodchikov equations and conformal blocks	p. 48
Braiding and fusing matrices	p. 50
Projective representation of mapping class group	p. 53
Three-dimensional manifold invariants via Heegard decomposition	p. 57
Calculations by topological quantum field theory	p. 59
Atiyah's axioms	p. 59
An example: connected sum	p. 60
Jones polynomials	p. 60
Surgery	p. 61
Verlinde's conjecture and its proof	p. 63
A brief survey on quantum group method	p. 64
Algebraic representation of knot	p. 64
Hopf algebra and quantum groups	p. 67
Chern-Simons theory and quantum groups	p. 68
Renormalized Perturbation Series of Chern-Simons-Witten Theory	p. 71
Path integral and morphism of Hilbert spaces	p. 71
One-dimensional quantum field theory	p. 71
Schroedinger operator	p. 72
Spectrum and determinant	p. 75
Asymptotic expansion and Feynman diagrams	p. 77
Asymptotic expansion of integrals, finite dimensional case	p. 77
Integration on a sub-variety	p. 81
Partition function and topological invariants	p. 82
Gauge fixing and Faddeev-Popov ghosts	p. 83
The leading term	p. 85
Wilson line and link invariants	p. 88
A brief introduction on renormalization of Chern-Simons theory	p. 89
A regulization scheme	p. 90
The Feynman rules	p. 91
Topological Sigma Model and Localization	p. 95
Constructing knot invariants from open string theory	p. 95
Introduction	p. 95
A topological sigma model	p. 96
Localization principle	p. 97
Large N expansion of Chern-Simons gauge theory	p. 98
Equivariant cohomology and localization	p. 99
Equivariant cohomology	p. 99
Localization, finite dimensional case	p. 100
Atiyah-Bott's residue formula and Duistermaat-Heckman formula	p. 101
Complex case, Atiyah-Bott's residue formula	p. 101
Symplectic case, Duistermaat-Heckman formula	p. 102
2D Yang-Mills theory by localization principle	p. 104
Cohomological Yang-Mills field theory	p. 104
Relation with physical Yang-Mills theory	p. 105
Evaluation of Yang-Mills theory	p. 107
Combinatorial approach to 2D Yang-Mills theory	p. 110
Complex Manifold Without Potential Theory	p. 113
Geometric Quantization of Chern-Simons Gauge Theory	p. 121
On Holomorphic Factorization of WZW and Coset Models	p. 169
Bibliography	p. 193
Index	p. 197
Afterwards	p. 199
Table of Contents provided by Syndetics. All Rights Reserved.

Lecture Notes On Chern-simons-witten Theory

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