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Cambridge Lectures on Supersymmetry and Extra Dimensions

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DAMTP-2010-90 IC/2010/xx MPP-2010-143 Cambridge Lectures on Supersymmetry and Extra Dimensions Lectures by: Fernando Quevedo1;2, Notes by: Sven Krippendorf1, Oliver Schlotterer3 1 DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK. 2 ICTP, Strada Costiera 11, Trieste 34151, Italy. 3 Max-Planck-Institut f¨urPhysik, F¨ohringerRing 6, 80805 M¨unchen,Germany. November 8, 2010 arXiv:1011.1491v1 [hep-th] 5 Nov 2010 2 Contents 1 Physical Motivation for supersymmetry and extra dimensions 9 1.1 Basic Theory: QFT . .9 1.2 Basic Principle: Symmetry . 10 1.2.1 Classes of symmetries . 10 1.2.2 Importance of symmetries . 10 1.3 Basic example: The Standard Model . 12 1.4 Problems of the Standard Model . 13 2 Supersymmetry algebra and representations 17 2.1 Poincar´esymmetry and spinors . 17 2.1.1 Properties of the Lorentz group . 17 2.1.2 Representations and invariant tensors of SL(2; C)..................... 18 2.1.3 Generators of SL(2; C).................................... 20 2.1.4 Products of Weyl spinors . 20 2.2 Supersymmetry algebra . 21 2.2.1 History of supersymmetry . 21 2.2.2 Graded algebra . 22 2.2.3 Representations of the Poincar´egroup . 24 2.2.4 N = 1 supersymmetry representations . 26 2.2.5 Massless supermultiplet . 26 2.2.6 Massive supermultiplet . 27 2.3 Extended supersymmetry . 29 2.3.1 Algebra of extended supersymmetry . 29 2.3.2 Massless representations of N > 1 supersymmetry . 30 2.3.3 Massive representations of N > 1 supersymmetry and BPS states . 32 3 Superfields and superspace 35 3.1 Basics about superspace . 35 3.1.1 Groups and cosets . 35 3.1.2 Properties of Grassmann variables . 37 3 4 CONTENTS 3.1.3 Definition and transformation of the general scalar superfield . 38 3.1.4 Remarks on superfields . 39 3.2 Chiral superfields . 40 3.3 Vector superfields . 41 3.3.1 Definition and transformation of the vector superfield . 41 3.3.2 Wess Zumino gauge . 42 3.3.3 Abelian field strength superfield . 42 3.3.4 Non - abelian field strength . 43 4 Four dimensional supersymmetric Lagrangians 45 4.1 N = 1 global supersymmetry . 45 4.1.1 Chiral superfield Lagrangian . 45 4.1.2 Miraculous cancellations in detail . 48 4.1.3 Abelian vector superfield Lagrangian . 49 4.1.4 Action as a superspace integral . 51 4.2 Non-renormalization theorems . 52 4.2.1 History . 52 4.2.2 Proof of the non-renormalization theorem . 53 4.3 N = 2; 4 global supersymmetry . 55 4.3.1 N =2............................................. 56 4.3.2 N =4............................................. 57 4.3.3 Aside on couplings . 57 4.4 Supergravity: an Overview . 58 4.4.1 Supergravity as a gauge theory . 58 4.4.2 The linear supergravity multiplet with global supersymmetry . 59 4.4.3 The supergravity multiplet with local supersymmetry . 60 4.4.4 N = 1 Supergravity in Superspace . 61 4.4.5 N = 1 supergravity coupled to matter . 61 5 Supersymmetry breaking 65 5.1 Basics . 65 5.2 F - and D breaking . 66 5.2.1 F term breaking . 66 5.2.2 O'Raifertaigh model . 66 5.2.3 D term breaking . 69 5.3 Supersymmetry breaking in N = 1 supergravity . 70 6 The MSSM - basic ingredients 71 6.1 Particles . 71 6.2 Interactions . 71 CONTENTS 5 6.3 Supersymmetry breaking in the MSSM . 73 6.4 The hierarchy problem . 74 6.5 The cosmological constant problem . 76 7 Extra dimensions 77 7.1 Basics of Kaluza Klein theories . 77 7.1.1 History . 77 7.1.2 Scalar field in 5 dimensions . 78 7.1.3 Vector fields in 5 dimensions . 79 7.1.4 Duality and antisymmetric tensor fields . 81 7.1.5 Gravitation in Kaluza Klein theory . 83 7.2 The brane world scenario . 85 7.2.1 Large extra dimensions . 87 7.2.2 Warped compactifications . 87 7.2.3 Brane world scenarios and the hierarchy problem . 89 8 Supersymmetry in higher dimensions 91 8.1 Spinors in higher dimensions . 91 8.1.1 Spinor representations in even dimensions D = 2n .................... 91 8.1.2 Spinor epresentations in odd dimensions D = 2n +1................... 92 8.1.3 Majorana spinors . 93 8.2 Supersymmetry algebra . 93 8.2.1 Representations of supersymmetry algebra in higher dimensions . 94 8.3 Dimensional Reduction . 96 8.4 Summary . 97 A Useful spinor identities 99 A.1 Bispinors . 99 A.2 Sigma matrices . 100 A.3 Bispinors involving sigma matrices . 100 B Dirac spinors versus Weyl spinors 103 B.1 Basics . 103 B.2 Gamma matrix technology . 104 B.3 The Supersymmetry algebra in Dirac notation . 105 C Solutions to the exercises 107 C.1 Chapter 2 . 107 C.2 Chapter 3 . 110 C.3 Chapter 4 . 114 C.4 Chapter 5 . 117 6 CONTENTS C.5 Chapter 7 . ..
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