Multiplicity diagrams can be viewed as schemes for describing the phenomenon of "symmetry breaking" in quantum physics. The subject of this book is the multiplicity diagrams associated with the classical groups U(n), O(n), etc. It presents such topics as asymptotic distributions of multiplicities, hierarchical patterns in multiplicity diagrams, lacunae, and the multiplicity diagrams of the rank 2 and rank 3 groups. The authors take a novel approach, using the techniques of symplectic geometry. The book develops in detail some themes which were touched on in the highly successful Symplectic Techniques in Physics by V. Guillemin and S. Sternberg (CUP, 1984) , including the geometry of the moment map, the Duistermaat-Heckman theorem, the interplay between coadjoint orbits and representation theory, and quantization. Students and researchers in geometry and mathematical physics will find this book fascinating.
Author(s): Victor Guillemin, Eugene Lerman, Shlomo Sternberg
Edition: 1
Publisher: Cambridge University Press
Year: 2009
Language: English
Pages: 238
Contents ......Page 7
Acknowledgments ......Page 11
Introduction ......Page 13
1.1 What Is a Symplectic Fiber Bundle? ......Page 17
1.2 Symplectic Connections ......Page 18
1.3 Minimal Coupling ......Page 21
1.4 The Coupling Form wr ......Page 23
1.5 Weak Coupling ......Page 28
1.6 Varying the Connection ......Page 33
2 Examples of Symplectic Fibrations: The Coadjoint Orbit Hierarchy ......Page 37
2.1 Basic Example ......Page 38
2.2 A Normal Form Theorem ......Page 43
2.3 Fibrations of Coadjoint Orbits and the Cross Section Theorem ......Page 45
2.4 Symplectic Mackey-Wigner Theory ......Page 61
2.5 Riemannian Submersions with Totally Geodesic Fibers ......Page 62
2.6 The Hannay-Berry Connection ......Page 67
3.1 Orbital Methods in Representation Theory ......Page 70
3.2 Computing the D-H Polynomials: The Case of Linear Action ......Page 81
3.3 Computing the D-H Polynomials: The "Heckman" Formula ......Page 92
3.4 Is There a "Kostant" Formula Corresponding to the Heckman Formula? ......Page 100
3.5 Computing the D-H Polynomials: Formulas for the Jumps ......Page 111
Appendix 3.A: The Duistermaat-Heckman Measure as the Volume of Reduced Spaces ......Page 117
Appendix 3.B: Localization and the Duistennaat-Heckman Formula ......Page 118
4.1 A Few Words about the Contents of This Chapter ......Page 123
4.2 The Heckman Formula ......Page 127
4.3 An Inductive Formula for the D-H Measure Associated with a Coadjoint Orbit of U(n) ......Page 132
4.4 The Kostant Formula ......Page 135
4.5 The Weak Coupling Limit ......Page 138
4.6 Reduction and Weak Coupling ......Page 141
4.7 Some Final Comments about Weak Coupling ......Page 146
5.1 Generalities about Toral Moment Maps ......Page 156
5.2 Singular Values of the Moment Map 4>: 0 -> t ......Page 157
5.3 Examples: SU(4) Orbits ......Page 162
5.4 Duistermaat-Heckman Polynomials for 10-Dimensional Orbit of SU(4) ......Page 169
5.5 Variation of Orbits ......Page 179
A.1 Weyl, Kostant, and Steinberg Formulas ......Page 183
A.2 The Action of the Casimir Element on Z()) ......Page 186
A.3 Determining the Coefficients ......Page 188
A.4 Proof of the Kostant and Weyl Formulas ......Page 191
B.1 Hidden Symmetries ......Page 193
B.2 Equivariant Cohomology ......Page 194
B.3 Superalgebras ......Page 196
B.4 Differential G Complexes ......Page 197
B.5 The Weil Algebra ......Page 200
B.6 The Matthai-Quillen Isomorphism ......Page 202
B.7 The Cartan Model ......Page 204
B.8 Locally Free Complexes ......Page 205
B.9 Equivariant Cohomology: Homogeneous Spaces ......Page 206
B.10 The Thorn Form According to Mattai-Quillen ......Page 211
B.11 Equivariant Superconnections ......Page 213
B.12 Equivariant Characteristic Classes ......Page 215
B.13 Reduction Formula for Locally Free Torus Actions ......Page 217
B.14 Localization ......Page 220
B.15 Localization in Stages ......Page 226
C. 1 Commutativity of Quantization and Reduction ......Page 227
C.3 Orbifolds ......Page 229
C.5 Nonisolated Fixed Points ......Page 230
Bibliography ......Page 231
Index ......Page 237
