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Anomaly-Free Supergravities in Six Dimensions Ph.D. Thesis arXiv:hep-th/0611133v1 12 Nov 2006 Spyros D. Avramis National Technical University of Athens School of Applied Mathematics and Natural Sciences Department of Physics Spyros D. Avramis Anomaly-Free Supergravities in Six Dimensions Dissertation submitted to the Department of Physics of the National Technical University of Athens in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics. Thesis Advisor: Alex Kehagias Thesis Committee: Alex Kehagias Elias Kiritsis George Zoupanos K. Anagnostopoulos A.B. Lahanas E. Papantonopoulos N.D. Tracas Athens, February 2006 Abstract This thesis reviews minimal N = 2 chiral supergravities coupled to matter in six dimensions with emphasis on anomaly cancellation. In general, six-dimensional chiral supergravities suffer from gravitational, gauge and mixed anomalies which, being associated with the breakdown of local gauge symmetries, render the theories inconsistent at the quantum level. Consistency of the theory is restored if the anomalies of the theory cancel via the Green-Schwarz mechanism or generalizations thereof, in a similar manner as in the case of ten-dimensional N = 1 supergravi- ties. The anomaly cancellation conditions translate into a certain set of constraints for the gauge group of the theory as well as on its matter content. For the case of ungauged theories these constraints admit numerous solutions but, in the case of gauged theories, the allowed solutions are remarkably few. In this thesis, we examine these anomaly cancellation conditions in detail and we present all solutions to these conditions under certain restrictions on the allowed gauge groups and representations, imposed for practical reasons. The central result of this thesis is the existence of two more anomaly-free gauged models in addition to the only one that was known until recently. We also examine anomaly cancellation in the context of Hoˇrava-Witten–type compactifications of 1 minimal seven-dimensional supergravity on S /Z2. Finally, we discuss some basic aspects of the 4D phenomenology of the gauged models. i Acknowledgements I thank my advisor Prof. Alex Kehagias for his guidance during my Ph.D. years. I would also like to thank the members of my advisory committee Profs. Elias Kiritsis and George Zoupanos and the members of my evaluation committee Profs. K. Anagnostopoulos, A.B. Lahanas, E. Papantonopoulos and N.D. Tracas. I am also grateful to all my teachers during my M.Sc. and Ph.D. time. I thank Prof. Seif Randjbar-Daemi for sharing with me his expertise on six-dimensional super- gravity, for inviting me to ICTP, and for a very interesting collaboration. I thank the Theory Division of CERN, the High Energy, Cosmology and Astroparticle Physics Section of ICTP and the Centre de Physique Th´eorique of the Ecole´ Polytechnique for hospitality during various stages of this work. I thank my colleague Constantina Mattheopoulou for our various discussions, for help with bibliography and for proofreading my thesis. I also thank my colleague and collaborator Dimitrios Zoakos for useful discussions. This thesis was financially supported from the “Heraclitus” program cofunded by the Greek State and the European Union. Finally, I have to thank my parents Dimitrios and Olympia and my brother Alexandros for their constant support throughout these years. iii Contents 1 Introduction 1 1.1 StringTheoryandSupergravity. ........ 1 1.2 Six-dimensionalSupergravity . ......... 2 1.3 AnomalyCancellation . .. .. .. .. .. .. .. .. ..... 4 1.4 OutlineoftheThesis.............................. ..... 6 2 Supersymmetry and Supergravity 7 2.1 Supersymmetry................................... ... 7 2.1.1 Thesupersymmetryalgebra . .... 8 2.1.2 Massless representations . ...... 11 2.2 Supergravity .................................... 17 2.2.1 Gauge theory of the super-Poincar´egroup . ......... 18 2.2.2 Puregravity ................................... 20 2.2.3 Simplesupergravity . .. .. .. .. .. .. .. .. 21 2.3 ScalarCosetManifolds. ...... 25 2.3.1 Coset spaces and their geometry . ..... 25 2.3.2 Nonlinear sigma models and gauging . ...... 27 2.4 Supergravities in D = 11 and in D =10 ........................ 29 2.4.1 Eleven-dimensional supergravity . ........ 29 2.4.2 Ten-dimensional N =1supergravity ...................... 30 3 Anomalies 33 3.1 IntroductiontoAnomalies. ....... 33 3.1.1 Singletanomalies.............................. 34 3.1.2 Nonabelian anomalies . 37 3.1.3 Physical aspects of anomalies . ...... 43 3.2 Calculation of Singlet Anomalies . ......... 45 3.2.1 The spin–1/2anomaly.............................. 45 3.2.2 The spin–3/2anomaly.............................. 50 v 3.2.3 The self-dual (n 1)–formanomaly ...................... 50 − 3.3 Calculation of Nonabelian Anomalies . .......... 51 3.3.1 The spin–1/2anomaly.............................. 51 3.3.2 The spin–3/2anomaly.............................. 53 3.3.3 The self-dual (n 1)–formanomaly ...................... 54 − 3.3.4 The consistent anomaly from the descent equations . ........... 54 3.4 TheAnomalyPolynomials. ..... 56 3.5 AnomalyCancellation . .. .. .. .. .. .. .. .. ..... 58 3.5.1 TheGreen-Schwarzmechanism . 58 3.6 Globalanomalies ................................. 60 3.6.1 Witten’s SU(2) anomaly . 61 3.6.2 Generalizations ............................... 63 4 Anomaly Cancellation in Superstring Theory and M-Theory 65 4.1 Anomaly Cancellation in Superstring Theory . ........... 65 4.1.1 Anomaly cancellation in Type IIB supergravity . .......... 65 4.1.2 Green-Schwarz anomaly cancellation in heterotic/TypeItheory. 66 4.2 Anomaly Cancellation in Heterotic M-theory . ........... 70 1 4.2.1 11–dimensional supergravity on S /Z2 ..................... 70 4.2.2 Anomalyanalysis............................... 72 4.2.3 Theboundaryaction. .. .. .. .. .. .. .. .. 75 4.2.4 Anomaly cancellation . 78 5 Anomaly-Free Supergravities in Six Dimensions 81 5.1 D = 6, N =2Supergravity ............................... 81 5.1.1 A quick survey of D =6supergravities. .. .. .. .. .. .. 81 5.1.2 General facts about D = 6, N =2supergravity . 82 5.1.3 Gravity and tensor multiplets . ...... 84 5.1.4 Vectormultiplets .............................. 84 5.1.5 Hypermultiplets ............................... 86 5.1.6 ThecompleteLagrangian . 93 5.2 Anomaly Cancellation in D = 6, N =2Supergravity. .. .. .. .. .. 94 5.2.1 Localanomalies................................ 94 5.2.2 Globalanomalies ............................... 100 5.3 KnownAnomaly-FreeModels . 101 5.3.1 Ungauged theories: Heterotic string theory on K3 .............. 101 5.3.2 Gauged theories: The E E U(1)model ................. 106 7 × 6 × 5.4 Searching for Anomaly-Free Theories . .......... 107 5.5 Anomaly-free Ungauged Supergravities . ........... 109 5.5.1 Simplegroups.................................. 109 5.5.2 Productsoftwosimplegroups . 113 5.6 Anomaly-free Gauged Supergravities . .......... 121 1 5.7 Anomaly Cancellation in D = 7, N = 2 Supergravity on S /Z2 ........... 124 5.7.1 Basics of D = 7, N =2supergravity . .. .. .. .. .. .. 125 1 5.7.2 Compactification on S /Z2 andanomalies .. .. .. .. .. .. 126 5.7.3 Anomaly cancellation . 128 5.8 Discussion...................................... 130 6 Magnetic Compactifications 133 6.1 Equations of Motion, Supersymmetry and the Scalar Potential . 133 6.2 Ungauged Theories: Compactification on a Magnetized Torus............ 136 6.3 Gauged Theories: Compactification on a Magnetized Sphere............. 138 6.3.1 TheSalam-Sezginmodel. 139 6.3.2 Generalembeddings . .. .. .. .. .. .. .. .. 141 6.3.3 The E E U(1) model .......................... 144 7 × 6 × R 6.3.4 The E G U(1) model .......................... 145 7 × 2 × R A Notation and conventions 147 B Gamma Matrices and Spinors in Diverse Dimensions 149 B.1 Gamma Matrices in Diverse Dimensions . ........ 149 B.1.1 TheCliffordalgebra ............................. 149 B.1.2 Gamma matrix identities . 150 B.2 Representations of the Clifford Algebra . .......... 151 B.2.1 Spinors of SO(D 1, 1) ............................. 151 − B.2.2 TheWeylandMajoranaconditions. 152 B.3 Six-DimensionalSpinors . ....... 156 C Characteristic Classes and Index Theorems 159 C.1 CharacteristicClasses . ....... 159 C.1.1 Generalities.................................. 159 C.1.2 Characteristic classes of complex vector bundles . ............ 161 C.1.3 Characteristic classes of real vector bundles . ............ 161 C.2 TheDescentEquations. 163 C.3 IndexTheorems................................... 165 D Anomaly Coefficients 167 D.1 SecondandFourthIndices. 167 D.2 AnomalyCoefficients............................... 167 Chapter 1 Introduction 1.1 String Theory and Supergravity Despite the impressive successes of the Standard Model of elementary particle physics, there still remain many fundamental problems unanswered up to date. The most important such problem is finding an appropriate quantum-mechanical formulation of gravity since, from the usual viewpoint of quantum field theory, the gravitational interaction is non-renormalizable. Another problem, of more theoretical interest, is finding a unified description of all forces in Nature. Up to now, the most successful attempts to solve these problems have been made within the framework of string theory [1, 2, 3, 4]. String theory was initially developed with the hope that it would explain certain properties of the strong interactions. Namely, the soft behavior of high-energy hadronic scattering amplitudes in the Regge limit could be approximated
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