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Black Holes in N = 2 Supergravity Harold Erbin

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Black Holes in N = 2 Supergravity Harold Erbin Black holes in N = 2 supergravity Harold Erbin To cite this version: Harold Erbin. Black holes in N = 2 supergravity. High Energy Physics - Theory [hep-th]. Université Pierre et Marie Curie - Paris VI, 2015. English. NNT : 2015PA066367. tel-01269849 HAL Id: tel-01269849 https://tel.archives-ouvertes.fr/tel-01269849 Submitted on 5 Feb 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Thèse de doctorat réalisée au Laboratoire de physique théorique et hautes énergies pour obtenir le grade de Docteur de l’Université Pierre et Marie Curie Spécialité : Physique École doctorale : « Physique en Île-de-France » présentée par Harold Erbin soutenue le 23 septembre 2015 Titre : Trous noirs en supergravité N = 2 Black holes in N = 2 supergravity devant le jury composé de Iosif Bena Membre invité Geoffrey Compère Rapporteur Atish Dabholkar Examinateur Nick Halmagyi Directeur de thèse Giuseppe Policastro Examinateur Henning Samtleben Rapporteur Antoine Van Proeyen Examinateur Contents Abstract.......................................... vii Résumé .......................................... vii Remerciements ...................................... viii Introduction ix 0.1 Background..................................... ix 0.1.1 Quantumgravityandstringtheory. .. ix 0.1.2 Supersymmetryandsupergravity . .. x 0.1.3 Blackholes ................................. xi 0.1.4 BPSsolutionsandadSblackholes . xiii 0.1.5 Taub–NUTspacetime . .. .. .. .. .. .. .. .. .. .. .. xiii 0.2 Motivations ..................................... xiv 0.2.1 Supergravity ................................ xiv 0.2.2 Demiański–Janis–Newmanalgorithm . ... xv 0.3 Content ....................................... xvii 0.3.1 Supergravity ................................ xvii 0.3.2 Demiański–Janis–Newmanalgorithm . ... xix 0.4 Structure ...................................... xx 0.5 Publications..................................... xxi 2I N = supergravity 1 1 Introduction to N = 2 supergravity 2 1.1 Algebraandmultiplets. .. 2 1.2 Lagrangian ..................................... 4 1.3 Electromagneticduality . ... 4 1.4 Scalargeometry................................... 7 1.4.1 Isometries.................................. 7 1.4.2 SpecialKählermanifolds . 8 1.4.3 Quaternionicmanifolds . 9 2 Gauged supergravity 11 2.1 Generalities ..................................... 11 2.2 Gaugings ...................................... 12 2.2.1 Isometries.................................. 12 2.2.2 Fayet–Iliopoulosgauging. .. 14 2.3 Lagrangian ..................................... 15 2.3.1 Generalcase ................................ 15 2.3.2 Fayet-Iliopoulosgaugings . .. 16 2.3.3 Minimalgaugedsugra . .. .. .. .. .. .. .. .. .. .. .. 16 2.4 Supersymmetryvariations . ... 17 2.5 Magneticgaugings ................................. 17 i 2.5.1 Generalities................................. 18 2.5.2 Constraintsfromlocality . 18 2.6 Quaternionicgaugings . .. 19 II Kähler geometries 21 3 Hermitian and Kähler manifolds 22 3.1 Hermitianmanifold................................ 22 3.1.1 Definitionandproperties . 22 3.1.2 Complexcoordinates. 23 3.2 Kählermanifold................................... 24 3.2.1 Definition .................................. 24 3.2.2 Riemanniangeometry . 25 3.2.3 Symmetries ................................. 25 3.2.4 Kähler–Hodgemanifold . 27 4 Special Kähler geometry 29 4.1 Definition ...................................... 29 4.2 Homogeneous coordinates and symplectic structure . .......... 30 4.2.1 Vectors ................................... 30 4.2.2 MetricandKählerpotential. 31 4.2.3 Covariantholomorphicfields . 31 4.2.4 Prepotentialproperties . 32 4.3 Homogeneousmatrices.. .. .. .. .. .. .. .. .. .. .. .. .. 32 4.3.1 Hessianmatrix ............................... 32 4.3.2 Periodmatrix................................ 33 4.4 Symplecticmatrices ............................... 33 4.5 Structurecoefficients. .. .. .. .. .. .. .. .. .. .. .. .. .. 35 4.6 Riemanniangeometry ............................... 36 4.6.1 Generalproperties . .. .. .. .. .. .. .. .. .. .. .. .. 36 4.6.2 Symmetricspace .............................. 37 5 Symplectic transformations and invariants 38 5.1 Symplectictransformations . .... 38 5.1.1 Holomorphicsection . 38 5.1.2 Sectionandcoordinates . 39 5.2 Symplecticinvariants. ... 40 5.3 Dualityinvariants................................ .. 41 5.3.1 Generaldefinition ............................. 41 5.3.2 Quarticinvariant.............................. 42 5.3.3 Invarianttensor............................... 42 5.3.4 Freudenthalduality .. .. .. .. .. .. .. .. .. .. .. .. 43 5.4 Non-symmetricspaces .. .. .. .. .. .. .. .. .. .. .. .. .. 43 6 Special Kähler classification. Quadratic and cubic prepotentials 44 6.1 Classificationofspaces. .. ... 44 6.1.1 Symmetricspaces.............................. 44 6.1.2 Homogeneousspaces .. .. .. .. .. .. .. .. .. .. .. .. 45 6.2 Quadraticprepotential. ... 45 6.2.1 Generalformulas .............................. 46 6.2.2 Quarticandquadraticinvariants . ... 47 6.3 Cubicprepotential ................................ 48 6.3.1 Generalcase ................................ 48 ii 6.3.2 Genericsymmetricmodels. 49 6.3.3 Jordanalgebrasandquarticinvariant . .... 49 6.3.4 Non-symmetricspaces . 50 7 Special Kähler isometries 51 7.1 Generalcase..................................... 51 7.2 Cubicprepotential ................................ 52 7.2.1 Parameters ................................. 53 7.2.2 Liederivative................................ 53 7.2.3 Algebra ................................... 53 7.3 Quadraticprepotential. ... 54 7.3.1 Parameters ................................. 54 7.3.2 Liederivative................................ 54 8 Quaternionic geometry 55 8.1 Definitions...................................... 55 8.2 Geometry ...................................... 57 8.2.1 Vielbein................................... 57 8.2.2 Curvature.................................. 58 8.3 Symmetries ..................................... 58 8.4 Classificationofspaces. .. ... 60 8.5 Specialquaternionicmanifolds . ..... 61 8.5.1 Quaternionicmetricfromthec-map . .. 61 8.5.2 BasespecialKählermanifold . 61 8.5.3 Geometricalstructures. 62 9 Quaternionic isometries 63 9.1 Killingvectors ................................... 63 9.1.1 Dualitysymmetries. 63 9.1.2 Extrasymmetries.............................. 64 9.1.3 Hiddenvectors ............................... 65 9.1.4 Summary .................................. 67 9.2 Algebra ....................................... 68 9.3 Compensators.................................... 69 9.3.1 Dualitysymmetries. 70 9.3.2 Hiddensymmetries. .. .. .. .. .. .. .. .. .. .. .. .. 70 9.4 Prepotentials .................................... 71 III BPS equations for black holes 72 10 Generalities on AdS–NUT black holes 73 10.1Ansatz........................................ 73 10.2 Motivation: constant scalar black holes . ......... 73 10.2.1 Solution................................... 73 10.2.2 Rootstructureandsupersymmetry . ... 74 10.3 RootstructureandIRgeometry . ... 75 11 Static BPS equations 77 11.1Ansatz........................................ 77 11.2Equations ...................................... 78 11.3 Symplecticextension . ... 79 11.4 Symmetric withFIgaugings ......................... 80 Mv iii 12 Static BPS solutions 82 12.1 N = 2adS4 ..................................... 82 12.2 Near-horizon adS2 Σg . ............................. 85 12.3Generalsolution .................................× . 87 12.4Examples ...................................... 88 12.4.1 V5,2 . .................................... 89 12.4.2 SO(5) gauged supergravity on M3 . ................... 90 13 BPS AdS–NUT black holes 92 13.1Ansatz........................................ 92 13.2BPSequations ................................... 93 13.3 Symmetric v withFIgaugings ......................... 94 13.4Solutions ......................................M 95 13.4.1 Pairofdoubleroots .. .. .. .. .. .. .. .. .. .. .. .. 95 13.4.2 Singledoubleroot . .. .. .. .. .. .. .. .. .. .. .. .. 96 13.4.3 Fourindependentroots . 98 13.5Examples ...................................... 99 13.5.1 T 3 model .................................. 99 13.5.2 Constantscalarsolution . .. 100 13.5.3 F = X0X1 . ............................... 100 − IV Demiański–Janis–Newman algorithm 101 14 Janis–Newman algorithm 102 14.1 Originalprescription . .... 102 14.2 Giampieri’sprescription . ..... 104 14.3Validity ....................................... 106 14.4Gaugefield ..................................... 106 14.4.1 Giampieri’sprescription . ... 106 14.4.2 Newman’sprescription. 107 14.4.3 Keane’sprescription . 107 14.5Examples ...................................... 107 14.5.1 Flatspace.................................. 107 14.5.2 Kerr–Newmanblackhole . 108 15 Extended algorithm 110 15.1 Settinguptheansatz. .. .. .. .. .. .. .. .. .. .. .. ... .. 111 15.2 Demiański–Janis–Newmanalgorithm . ...... 112 15.2.1 Janis–Newmantransformation . .. 112 15.2.2 Metric.................................... 113 15.2.3 Gaugefield ................................
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