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On a unified description of non-abelian charges, monopoles and dyons
Kampmeijer, L.
Publication date
2009
Link to publication
Citation for published version (APA):
Kampmeijer, L. (2009). On a unified description of non-abelian charges, monopoles and
dyons.
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Contents
Preface vii
1 Introduction 1
1.1 Labelling of monopoles in non-abelian phases . . . 2
1.2 Dyonic complications and the skeleton group . . . 5
2 Classical monopole solutions 9 2.1 Singular monopoles . . . 9
2.2 BPS monopoles . . . 12
2.3 Magnetic charge lattices . . . 15
2.3.1 Quantisation condition for singular monopoles . . . 16
2.3.2 Quantisation condition for smooth monopoles . . . 19
2.3.3 Quantisation condition for smooth BPS monopoles . . . 22
2.3.4 Murray condition . . . 25
3 Fusion rules for smooth BPS monopoles 31 3.1 Generating charges . . . 32
3.1.1 Generators of the Murray cone . . . 33
3.1.2 Generators of the magnetic weight lattice . . . 37
3.1.3 Generators of the fundamental Weyl chamber . . . 38
3.1.4 Generators of the fundamental Murray cone . . . 43
3.2 Moduli spaces for smooth BPS monopoles . . . 46
3.2.1 Framed moduli spaces . . . 46
3.2.2 Parameter counting for abelian monopoles . . . 48
3.2.3 Parameter counting for non-abelian monopoles . . . 50
3.3 Fusion rules of non-abelian monopoles . . . 56
3.3.1 Patching smooth BPS solutions . . . 56
3.3.2 Murray cone vs fundamental Murray cone . . . 58
3.3.3 Patching singular BPS solutions . . . 61
3.3.4 Towards semi-classical fusion rules . . . 63 ix
4 The skeleton group as a unified framework 67
4.1 Lie algebra conventions . . . 68
4.2 Charge sectors of the theory . . . 69
4.2.1 Electric charge lattices . . . 70
4.2.2 Magnetic charge lattices . . . 70
4.2.3 Dyonic charge sectors . . . 71
4.3 Skeleton Group . . . 72
4.3.1 Semi-direct products . . . 73
4.3.2 Maximal torus and its dual . . . 73
4.3.3 Weyl group action . . . 75
4.3.4 Proto skeleton group . . . 76
4.3.5 Definition of the skeleton group . . . 76
4.4 Representation theory . . . 78
4.4.1 Representation theory for semi-direct products . . . 78
4.4.2 Weyl orbits and centraliser representations . . . 81
4.4.3 Representations of the skeleton group . . . 82
4.4.4 Fusion rules . . . 84
4.4.5 Fusion rules for the skeleton group of SU(2) . . . 85
4.5 S-duality . . . 89
4.5.1 S-duality for simple Lie groups . . . 89
4.5.2 S-duality on charge sectors . . . 91
4.5.3 S-duality and skeleton group representations . . . 91
4.6 Gauge Fixing and non-abelian phases . . . 93
4.6.1 The abelian gauge and the skeleton gauge . . . 94
4.6.2 Gauge singularities and gauge artifacts . . . 95
4.6.3 Generalised Alice phases . . . 99
4.6.4 Unified electric-magnetic descriptions . . . 100
4.6.5 Phase transitions: condensates and confinement . . . 102
A The algebra underlying the Murray cone 107 A.1 Truncated group algebra . . . 108
A.2 Representation theory . . . 111
A.3 Reconstructing a semi-group . . . 113
B Weyl groups 117 B.1 Weyl groups of classical Lie algebras . . . 117
B.2 Representations of the Weyl group . . . 120
C Proto skeleton groups for classical Lie groups 123 C.1 Proto skeleton group for SU(2) . . . 123
C.2 Proto skeleton group for SU(3) . . . 124 x
C.3 Proto skeleton group for Sp(4) . . . 126
D Skeleton groups for classical Lie groups 127 D.1 Skeleton group for SU(n) . . . 127
D.2 Skeleton group for Sp(2n) . . . 129
D.3 Skeleton group for SO(2n+1) . . . 130
D.4 Skeleton group for SO(2n) . . . 131
E Generalised transformation group algebras 133 E.1 Irreducible representations . . . 133
E.2 Matrix elements and characters . . . 135
E.3 Fusion rules . . . 137
Summary 147
Samenvatting 151
Dankwoord 155
