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Two-Dimensional Quantum Gravity Coupled to Non-Conformal Matter Corinne De Lacroix De Lavalette

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Two-Dimensional Quantum Gravity Coupled to Non-Conformal Matter Corinne De Lacroix De Lavalette Two-dimensional quantum gravity coupled to non-conformal matter Corinne de Lacroix de Lavalette To cite this version: Corinne de Lacroix de Lavalette. Two-dimensional quantum gravity coupled to non-conformal matter. Quantum Physics [quant-ph]. Université Pierre et Marie Curie - Paris VI, 2017. English. NNT : 2017PA066288. tel-01706737 HAL Id: tel-01706737 https://tel.archives-ouvertes.fr/tel-01706737 Submitted on 12 Feb 2018 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 pour obtenir le grade de Docteur de l’Université Pierre et Marie Curie Spécialité : Physique École doctorale : « Physique en Île-de-France » présentée par Corinne de Lacroix soutenue le 28 septembre 2017 Titre : Gravité quantique à deux dimensions couplée à de la matière non-conforme Two-dimensional quantum gravity coupled to non-conformal matter devant le jury composé de Adel Bilal Directeur de thèse Benoit Douçot Examinateur Semyon Klevtsov Examinateur Antti Kupiainen Rapporteur Stam Nicolis Rapporteur Contents Abstract.......................................... iv 1 Introduction 1 1.1 Whyquantumgravity?.............................. 1 1.2 Two-dimensionalquantumgravity . ..... 2 1.3 Other research directions during this PhD . ........ 4 1.3.1 A fermionic matrix model for black holes . .... 5 1.3.2 Stringfieldtheory ............................. 6 1.4 Outlineofthethesis .............................. 7 1.5 Publications.................................... 7 I Quantum gravity in two dimensions with massive matter 9 2 Classical two-dimensional gravity 10 2.1 Generalconsiderations . .... 10 2.2 Degreesoffreedom ................................ 12 2.3 Dynamicsofunitarymatter . 13 3 Spectral theory and heat kernel 15 3.1 LaplaciansonaRiemannsurface . .... 15 3.1.1 Geometric Laplacians and operators of Laplace type . ........ 15 3.1.2 Complex tensor calculus and generalized Laplacians . ......... 16 3.2 Spectralfunctions ............................... 19 3.2.1 Green’sfunctions.............................. 19 3.2.2 Theheatkernel............................... 20 3.2.3 Zetafunction ................................ 21 3.3 Determinantregularization . ..... 22 3.3.1 Thepathintegralformalism. 22 3.3.2 Zetafunctionregularization . .... 23 3.3.3 Heat kernel short time regularization procedure . ......... 24 3.3.4 Variation of determinants of complex Laplacians . ........ 25 4 Two-dimensional quantum gravity 28 4.1 Functionalintegral .............................. 28 4.1.1 Gaugefixing ................................ 28 4.1.2 Theghostcontribution. 32 4.1.3 Gravitationalaction . 33 4.2 The conformalcaseandthe Liouville action . ........ 34 4.2.1 Basics of conformal field theory in two dimensions . ....... 34 4.2.2 Theconformalanomaly . 37 4.2.3 Conformal deformations and DDK anstatz . .... 38 4.3 Kählerformalism................................. 42 i 4.3.1 Kählerparametrization . 42 4.3.2 Gravitationalfunctionals . 43 4.4 TheLiouvilleaction .............................. 44 4.4.1 Basicproperties .............................. 44 4.4.2 Fromvariabletofixedarea . 45 4.4.3 Changeoftheintegrationmeasure . 46 4.4.4 Statesandcorrelationfunctions. ..... 47 4.5 TheMabuchiaction ................................ 50 4.5.1 Duality between the Liouville and Mabuchi actions . ........ 50 4.5.2 Hamiltonian................................. 51 5 Massive matter on a Riemann surface with boundaries 54 5.1 Integrationonasurfacewithboundaries . ........ 54 5.1.1 Boundaryintegrals. .. .. .. .. .. .. .. .. .. .. .. .. 54 5.1.2 Boundaryconditions . 56 5.2 Thegravitationalaction . .... 56 5.2.1 Perturbationtheory .. .. .. .. .. .. .. .. .. .. .. .. 57 5.2.2 Variationofthedeterminant . 58 5.3 Green’s functions and heat kernel on a manifold with boundaries . 58 5.3.1 Examples .................................. 58 5.3.2 Theheatkernelcontinued . 62 5.4 Local zeta functions and Green’s function at coinciding points. 64 5.5 The Mabuchi action on a manifold with boundaries . ........ 68 (0) 5.5.1 Variation of GR,bulk ............................ 69 (0) 5.5.2 Variation of Gζ .............................. 71 5.6 Thecylinder.....................................e 72 5.6.1 Green’sfunctiononthetoruse . 72 5.6.2 Green’s function and Green’s functions at coinciding points on the cylinder ................................... 74 5.6.3 Laplacian of GR and Gζ onthecylinder . 75 5.6.4 The gravitational action on the cylinder . ...... 77 6 Gravitational action for massive fermionic matter 78 6.1 Two-dimensionalspinors . .... 78 6.1.1 Cliffordalgebraandgammamatrices. 78 6.1.2 DiracandMajoranaspinors. 79 6.1.3 Gammamatrixrepresentations . 79 6.2 Majoranafermionfieldtheory. .... 80 6.2.1 Action.................................... 80 6.2.2 Functionalintegral . 81 6.2.3 Spectralfunctions ............................. 82 6.3 Gravitationalaction . .. .. .. .. .. .. .. .. .. .. .. .. 83 6.3.1 Conformalvariations. 83 6.3.2 Perturbationtheory .. .. .. .. .. .. .. .. .. .. .. .. 84 6.3.3 Variationofthezetafunction . 84 6.3.4 Themasslesscase ............................. 87 6.3.5 Themassivecase.............................. 87 6.3.6 Outlook................................... 91 ii 7 Spectrum of the Mabuchi action from a minisuperspace analysis 92 7.1 RescalingoftheMabuchiaction . .... 92 7.2 Computations of the minisuperspace Hamiltonian . .......... 94 7.2.1 Minisuperspaceapproximation . 95 7.2.2 First derivation: infinite area and flat limits . ........ 96 7.2.3 Second derivation: Legendre transformation . ........ 98 7.2.4 Third derivation: Ostrogradskiformalism . ....... 99 7.3 Minisuperspace canonical quantization of the Mabuchi theory . .. .. .. 100 II A quantum mechanical model for black holes from hologra- phy 102 8 Black holes and holography 103 8.1 Thethermalbehaviourofblackholes. ...... 103 8.1.1 Thermodynamicsofblackholes . 103 8.1.2 Theinformationparadox . 105 8.2 Studying black holes from holography . ...... 105 8.2.1 TheAdS/CFTcorrespondence . 105 8.2.2 BlackholesinAdS/CFT. 107 8.2.3 Building quantum models for black holes from holography....... 108 9 A fermionic matrix model for black holes 110 9.1 Descriptionofthemodel. 110 9.2 ConstructionoftheHilbertspace . ...... 112 9.3 ComputationofHamiltonians . .... 113 9.4 Results........................................ 115 III Appendices 118 A Résumé en français 119 A.1 Nécessité d’une théorie de gravité quantique . .......... 119 A.2 Gravité quantique à deux dimensions . ...... 120 A.2.1 Motivations................................. 120 A.2.2 Jauge conforme et action gravitationnelle . ....... 122 A.2.3 Action gravitationnelle pour un champ scalaire massif sur une surface deRiemannavecbords .......................... 125 A.2.4 Action gravitationnelle pour un spineur de Majorana . ......... 126 A.2.5 Spectre de l’action de Mabuchi dans l’approximation du minisuperespace128 A.3 Autres directions de recherche explorées pendant cette thèse.. .. .. .. 129 A.3.1 Un modèle de matrices fermioniques pour décrire des trous noirs . 130 A.3.2 Théoriedeschampsdecordes. 133 Bibliography 134 iii Abstract Finding a theory of quantum gravity which describes in a consistent way the quantum properties of matter and spacetime geometry is one of the greatest challenges of modern theoretical physics. However after several decades of research it still looks like a wild territory and a lot of conceptual and technical issues need to be resolved. A glimpse of the properties such a theory should have can be granted by the study of simplified toy models that allow for exact computations. In this thesis we will take this approach from two different points of view. The first part deals with two-dimensional quantum gravity. In two dimensions quan- tum gravity is much better understood and many computations can be carried out exactly. Whereas two-dimensional quantum gravity coupled to conformal matter has been widely studied and is now well understood, much less was known until recently about what hap- pens when matter is non-conformal. This is the issue we will focus on in this part of the thesis. First we compute the gravitational action in two cases: a massive scalar field on a Riemann surface with boundaries and a massive Majorana fermion on a manifold without boundary. The last case corresponds to a CFT perturbed by a conformal perturbation and is usually tackled through the DDK ansatz. However as we will see the results do not seem to match. Finally we give a minisuperspace computation of the spectrum of the Mabuchi action, a functional that has been shown to appear in the gravitational action for a massive scalar field. In the second part we focus on black hole thermal behaviour which provides a lot of insight of how a theory of quantum gravity should look like. In the context of string theory the AdS/CFT correspondence provides powerful tools for understanding the microscopic origin of black holes thermodynamics. Here we will construct a quantum mechanical toy model based on holographic principles to study the dynamics of quantum black holes. iv Chapter
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