I Parallelizable manifolds 6

Contents

1 Introduction 1

I Parallelizable manifolds 6

2 Parallelizability 7

2.1 Absolute Parallelism . . . . 7

2.2 Division Algebras . . . . 8

3 Lorentz Flatness 11 3.1 Flatness of a (Pseudo-)Riemannian Manifold . . . 12

3.1.1 Three Dimensional Example . . . 14

3.1.2 A General Discussion . . . 16

3.2 Flat Connections in Seven Dimensions . . . 18

3.2.1 Euclidean Signature . . . 18

3.2.2 Pseudo-Riemannian Signature . . . 22

4 S

Reduction and Exotic Solutions 26 4.1 Duality and Unorthodox Spacetime Signatures . . . 26

4.2 Action Principle and General Ansatz . . . 29

4.3 Freund-Rubin Type Solutions . . . 31

4.4 Englert Type Solutions . . . 32

4.4.1 Breaking of Symmetry . . . 37

II BRST Cohomology 39 5 Electric-Magnetic Duality 40 5.1 Electric-magnetic duality in Maxwell theory . . . 40

5.2 Non-compact duality group . . . 42

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5.3 A toy model inspired by N = 4 supergravity in four dimensions . . 46

6 Equivalent description in terms of the Embedding tensor formal- ism 48 6.1 Gauged Supergravity . . . 48

6.2 Electric-Magnetic duality and Gauging . . . 51

6.3 Embedding Tensor . . . 53

6.4 Yang-Mills deformations of first-order action . . . 57

6.5 Transition to the second-order formalism . . . 59

6.5.1 Choice of symplectic frame and locality restriction . . . 60

6.5.2 Electric group . . . 63

6.6 Deformation leading to the embedding tensor formalism . . . 65

7 BV-BRST Formalism and Deformation 73 7.1 A quick look at BV-BRST formalism . . . 73

7.2 BRST cohomology . . . 74

7.3 Batalin-Vilkovisky antifield formalism . . . 76

7.3.1 Depth of an element . . . 80

7.4 Properties of BRST differential . . . 81

7.5 Deformation of the Embedding tensor formalism . . . 82

8 BRST cohomology of scalar-vector coupled models 86 8.1 Abelian vector-scalar models in 4 dimensions . . . 90

8.1.1 Structure of the models . . . 90

8.1.2 Consistent deformations . . . 92

8.1.2.1 Solutions of U -type (a

non trivial) . . . 93

8.1.2.2 Solutions of W and V -type (vanishing a

but a

non trivial) . . . 96

8.1.2.3 Solutions of I-type (vanishing a

and a

) . . . 97

8.1.3 Local BRST cohomology at other ghost numbers . . . 97

8.1.3.1 h-terms . . . 97

8.1.3.2 Explicit description of cohomology . . . 98

8.1.3.3 Depth of solutions . . . 101

8.2 Antibracket map and structure of symmetries . . . 102

8.2.1 Antibracket map in cohomology . . . 102

8.2.2 Structure of the global symmetry algebra . . . 103

8.2.3 Parametrization through symmetries . . . 107

8.2.4 2nd order constraints on deformations and gauge algebra . 108 8.3 Quadratic vector models . . . 109

8.3.1 Description of the model . . . 109

8.3.2 Constraints on U , W -type symmetries . . . 110

8.3.3 Electric symmetry algebra . . . 111

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8.3.4 Restricted first order deformations . . . 113

8.3.5 Complete restricted deformations . . . 114

8.3.6 Remarks on GL(n

) transformations . . . 115

8.3.7 Comparison with the embedding tensor constraints . . . 116

8.4 Applications . . . 117

8.4.1 Abelian gauge fields: U -type gauging . . . 117

8.4.2 Abelian gauge fields with uncoupled scalars: U, V -type gaug- ings . . . 118

8.4.3 Bosonic sector of N = 4 supergravity . . . 119

8.5 First order manifestly duality-invariant actions . . . 123

8.5.1 Non-minimal version with covariant gauge structure . . . . 123

8.5.2 Local BRST cohomology . . . 126

8.5.3 Constraints on W, U-type cohomology . . . 126

8.5.4 Remarks on GL(2n

) transformations . . . 128

8.5.5 Application to the bosonic sector of N = 4 supergravity . . 129

9 Conclusions and Prospects 130 A Notation 134 B Octonion and Split Octonion 136 B.1 The Cayley-Dickson Construction of Divison Algebras . . . 136

B.2 Octonions . . . 137

B.3 Split Octonions . . . 138

B.4 Proof of Theorem 2.2 . . . 140

C The Pseudo-Sphere S

and the Split Octonions 142 C.1 A Survey of S

as a Homogeneous Space . . . 142

C.2 Two Infinite Family of Parallelizations of the Pseudo-Sphere S

. 143 C.3 Spin

(3, 4) and G

. . . 144

D Derivation of Equation (8.82) 146

Bibliography 151

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