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Contents

Introduction 1

Notations and conventions 7

I Preliminary material 8

1 Symmetry breaking in quantum field theory 11

1.1 Spontaneous breaking and Goldstone theorem . . . 11

1.1.1 Relativistic Goldstone theorem . . . 13

1.1.2 Coleman theorem . . . 16

1.2 Explicit breaking and Ward identities . . . 19

1.3 Goldstone theorem in non-relativistic field theories . . . 23

1.3.1 Relativistic-like theories with chemical potential . . . 28

1.3.2 Lifshitz invariant field theories . . . 31

2 Holography from a boundary-oriented perspective 37 2.1 Holographic renormalization . . . 40

2.1.1 Two-point functions from the renormalized action . . . 46

2.1.2 Holographic renormalization in presence of logarithms . . . 49

II Symmetry breaking in holographic field theories 52 3 Symmetry breaking in relativistic holographic setups 55 3.1 Spontaneous and explicit breaking in a holographic-dual CFT . . . 55

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Contents

3.2 The 1+1 dimensional case: Coleman fate in holography . . . 67

3.2.1 Maxwell gauge field in AdS3 . . . 68

3.2.2 Holographic renormalization with a charged scalar. . . 71

3.2.3 Symmetry breaking and Goldstone boson in AdS3/CFT2 . . . 78

4 Holographic type B non-relativistic GB 83 4.1 Renormalized action for a holographic type B GB model . . . 84

4.2 Holographic WI’s for symmetry breaking in presence of charge density . . . 88

5 Symmetry breaking in Lifshitz holography 93 5.1 Two-point functions for Lifshitz scalars . . . 94

5.1.1 The hard wall solution for Lifshitz Klein-Gordon scalar. . . 97

5.1.2 Two-point function for a general scalar . . . 99

5.2 Lifshitz holography for a massless vector . . . 102

5.2.1 Two-point function for a Lifshitz vector . . . 106

5.3 Lifshitz holography for symmetry breaking . . . 108

5.3.1 Holographic renormalization and Ward identities. . . 111

Main results 115

Bibliography 119

Références

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