Table of contents
I
Gribov Ambiguity
5
1 Introduction to Part I 7 2 Constrained Systems 11 2.1 Singular Lagrangians . . . 11 2.2 Primary Constraints . . . 12 2.3 Secondary Constraints . . . 152.4 First and Second Class Constraints . . . 15
2.5 Gauge Transformations . . . 16
2.6 Gauge Conditions . . . 18
3 The Gribov Problem 21 3.1 Gauge Fixing . . . 21
3.2 Gribov Region . . . 23
3.2.1 Alternative Definition . . . 25
3.3 Semi-classical Gribov Approach to QCD . . . 26
4 Semi-Classical Gribov Approach at Finite Temperature 29 4.1 Dynamical Thermal Mass . . . 29
4.2 Thermal Gap Equation . . . 31
4.3 The Three Regimes . . . 33
4.3.1 High Temperature Running Coupling . . . 33
4.3.2 Infrared Continuation . . . 35
5 Gribov Ambiguity and Degenerate Systems 41 5.1 Degenerate Systems . . . 41
5.2 Gauge Fixing and Gribov Ambiguity . . . 44
5.3 Gribov Horizon and Degenerate Surfaces . . . 46
5.4 The FLPR Model . . . 48
5.4.1 Effective Lagrangian for the Gauge-fixed System . . . 53
viii Table of contents
5.4.3 Quantization . . . 55
5.5 Irregular Case . . . 56
5.5.1 Example: Christ-Lee Model . . . 58
6 Conclusions and Future Directions 61
II
Three-dimensional Gravity
65
7 Introduction to Part II 67 8 Classical Duals of Three-dimensional Gravity 69 8.1 Einstein Gravity in Three-dimensions . . . 698.1.1 Chern-Simons Formulation . . . 70
8.1.2 Lorentz Flat Geometries . . . 70
8.2 Classical Dual Field Theories . . . 73
8.2.1 AdS3Case . . . 73
8.2.2 Flat Case . . . 77
8.2.3 Lorentz-flat-Geometries . . . 79
9 Actions on Coadjoint orbits 83 9.1 Coadjoint Orbits . . . 83
9.2 Kirillov-Kostant Form and Geometrical Actions . . . 85
9.3 Hamiltonian Formulation . . . 86
10 Geometrical Actions for 3D Gravity 91 10.1 Central Extensions . . . 92
10.2 Geometrical Actions for Centrally Extended Groups . . . 94
10.2.1 Kac-Moody Group . . . 95
10.2.2 Virasoro Group . . . 97
10.3 Semi-direct Products . . . 99
10.3.1 Adjoint Representation . . . 100
10.4 Centrally Extended Semi-direct Products . . . 101
10.4.1 Kac-Moody Algebra of G ⋉ g . . . 102
10.4.2 BMS3 . . . 103
11 Conclusions and Future Developments 107
References 109
Table of contents ix
Appendix B Generalities on Lie Groups 119 Appendix C Details on Maurer-Cartan Forms 123
C.1 Kac-Moody Group . . . 123
C.2 Virasoro Group . . . 124
C.3 Kac-Moody Group of G ⋉ g . . . 125
