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Loop Quantisation of Supergravity Theories Schleifenquantisierung Loop quantisation of supergravity theories Schleifenquantisierung von Supergravitationstheorien Der Naturwissenschaftlichen Fakult¨at der Friedrich-Alexander-Universit¨atErlangen-N¨urnberg zur Erlangung des Doktorgrades Dr. rer. nat. vorgelegt von Norbert Bodendorfer aus Waldbr¨ol Als Dissertation genehmigt von der Naturwissen- schaftlichen Fakult¨atder Friedrich-Alexander Universit¨at Erlangen-N¨urnberg Tag der m¨undlichen Pr¨ufung: 30. April 2013 Vorsitzende/r der Promotionskommission: Prof. Dr. Johannes Barth Erstberichterstatter/in: Prof. Dr. Thomas Thiemann Zweitberichterstatter/in: Prof. Dr. Kristina Giesel Contents Table of contents 1 About this thesis 4 Abstract 5 Zusammenfassung (german abstract) 6 Notation 7 1 Introduction 8 2 Overview of the results 11 I Connection dynamics for classical higher-dimensional general relativity 14 3 Hints from the Palatini action 15 3.1 The Ashtekar-Barbero variables . 15 3.2 Canonical analysis of the higher-dimensional Palatini action . 16 3.3 Gauge unfixing . 20 3.3.1 Toy model . 20 3.3.2 General theory . 21 3.3.3 Application to gravity . 23 4 Canonical transformation 24 4.1 Phase space extension and canonical transformation . 24 4.2 The Hamiltonian constraint in the new variables . 28 5 The Linear simplicity constraint 30 5.1 Introduction . 30 5.2 Introducing linear simplicity constraints . 31 II Loop quantum gravity in higher dimensions 34 6 LQG-techniques in higher dimensions 35 6.1 Kinematical quantisation techniques . 35 6.1.1 Holonomies, fluxes, and right invariant vector fields . 35 6.1.2 Solution of the Gauß and spatial diffeomorphism constraints . 37 6.2 Regularisation of the Hamiltonian constraint operator . 38 1 6.2.1 Volume operator . 38 6.2.2 Poisson bracket identities and regularisation . 38 6.3 Regularisation of the simplicity constraint . 39 7 Hilbert space techniques for the linear simplicity constraint 42 8 The simplicity constraint 45 8.1 Introduction . 45 8.2 The quadratic simplicity constraint operators . 47 8.2.1 A maximal closing subset of vertex constraints . 47 8.2.2 The solution space of the maximal closing subset . 49 8.2.3 Remarks . 50 8.3 The linear simplicity constraint operators . 53 8.3.1 Regularisation and anomaly freedom . 53 8.3.2 Solution on the vertices . 55 8.3.3 Edge constraints . 56 8.3.4 Remarks . 57 8.3.5 Mixed quantisation . 60 8.4 Comparison to existing approaches . 61 8.4.1 Continuum vs. discrete starting point . 62 8.4.2 Projected spin networks . 62 8.4.3 EPRL model . 64 8.5 Discussion and conclusions . 65 III Extensions to supergravity 67 9 Standard matter 68 10 Rarita-Schwinger field 70 10.1 Introduction . 70 10.2 Review of canonical supergravity . 72 10.2.1 Status of canonical supergravity . 72 10.2.2 Canonical supergravity in the time gauge . 73 10.3 Phase space extension . 75 10.3.1 Symplectic structure in the SO(D) theory . 75 10.3.2 SO(D + 1) gauge supergravity theory . 79 10.4 Background independent Hilbert space representations for Majorana fermions . 86 10.5 Generalisations to different multiplets . 90 10.5.1 Majorana spin 1=2 fermions . 90 10.5.2 Mostly plus / mostly minus conventions . 91 10.5.3 Weyl fermions . 91 10.6 Conclusions . 92 11 p-form gauge fields 94 11.1 Introduction . 94 11.2 Classical Hamiltonian analysis of the 3-index-photon action . 95 11.3 Reduced phase space quantisation . 99 11.4 Conclusions . 104 2 IV Initial value quantisation of higher-dimensional general relativity 105 12 Loop quantum gravity without the Hamiltonian constraint 106 12.1 Introduction . 106 12.2 Classical analysis . 108 12.3 Quantisation . 111 12.4 Geometric operators . 111 12.5 Application to black hole entropy . 112 12.6 Concluding remarks . 112 V Isolated horizon boundaries in higher-dimensional LQG 114 13 Classical phase space description of isolated horizon boundaries 115 13.1 Introduction . 115 13.2 General strategy . 116 13.3 Higher-dimensional isolated horizons and Lagrangian framework . 118 13.4 SO(D + 1) as internal gauge group . 120 13.5 Inclusion of distortion . 122 13.5.1 Beetle-Engle method . 122 13.5.2 Perez-Pranzetti method . 123 13.6 Comments on quantisation . 124 13.7 Concluding remarks . 125 VI Conclusion 127 14 Concluding remarks and further research 128 14.1 Summary of the results . 128 14.2 Towards loop quantum supergravity: Where do we stand? . 129 14.3 Further research . 130 A Simple irreps of SO(D + 1) and square integrable functions on the sphere SD 134 Danksagung (german acknowledgements) 136 Bibliography 137 3 About this thesis This thesis has grown out of collaborations with Alexander Stottmeister, Thomas Thiemann, and Andreas Thurn. The work started during my diploma thesis in February 2009, while the official starting date is December 2009, when I obtained my physics diploma. The topic of this thesis has been proposed by my supervisor Prof. Dr. Thomas Thiemann. The work for my diploma thesis, which contained parts of chapter 3, has been mainly conducted at the Albert-Einstein-Institute in Potsdam, which I thank for hospitality. Also during my diploma thesis, I have been supported by the Friedrich-Naumann-Foundation, the Max-Weber- Programme of Bavaria, the Leonardo-Kolleg of the University of Erlangen-N¨urnberg, the Elite Network Bavaria, and e-Fellows. My undergraduate and doctoral studies have been conducted in the Physics Advanced programme of the Universities Erlangen-N¨urnberg and Regensburg. The research which lead to this thesis has been performed at the Institute for Theoretical Physics III of the University of Erlangen-N¨urnberg. I strongly acknowledge financial and ideational support of the German National Merit Foundation. Also, I acknowledge support of the Elite Network Bavaria and e-Fellows. 4 Abstract The aim of this thesis is to develop loop quantisation techniques for higher-dimensional supergravities in order to make progress towards comparing loop quantum gravity to superstring theory. The abstract idea for this comparison is to look at loop quantum gravity and string theory as two methods of quantising an underlying gravity theory. While loop quantum gravity circumvents the usual problems associated with standard Fock-type quantisations of gravity theories by starting classically from a different Poisson-*-subalgebra of phase space functions, string theory introduces the string as a fundamental ingredient to cope with the short distance divergences arising in perturbative quantum gravity. The most obvious problem with comparing these two approaches to quantum gravity is that while it was only known how to formulate loop quantum gravity in four (or three) dimensions, including all types of standard model matter couplings, string theory requires a ten-dimensional spacetime with supersymmetry. The possible reductions of string theory to four dimensions on the other hand are highly non-unique, thus it is very hard to compare these approaches at this level. A possible solution to this problem, which is investigated in this thesis, is to extend the techniques of loop quantum gravity to higher-dimensional supergravities, which arise as the low energy limits of string theories. These supergravities have the same low energy particle content and live in the same spacetime dimension as the corresponding string theories and are thus suited to circumvent the above problem. Since this solution does not seem to be overly sophisticated at a first glance, it is important to remark that, prior to recent work on which this thesis is based, it was not even known how to formulate loop quantum gravity for pure higher-dimensional gravity, since the main ingredient at the classical level, the Ashtekar-Barbero connection, is only available in four dimensions. In this thesis, we will first develop a generalisation of this type of connection formulation for higher-dimensional general relativity. We will then show that the quantisation techniques from loop quantum gravity can be mainly carried over to this new formulation, while some new ingredients have to be introduced. Next, we are going to extend this formulation to higher-dimensional supergravities, thus providing a solution for the above problem. On a different route, we provide a partially reduced phase space quantisation of higher-dimensional general relativity conformally coupled to a scalar field based on the constant mean curvature gauge in the fourth part of this thesis. Finally, in the fifth part of this thesis, we will treat isolated horizons as boundaries of the spacetime manifold in order to take first steps to compare string theory and higher-dimensional loop quantum gravity in the presence of a black hole. 5 Zusammenfassung Es ist das Ziel dieser Arbeit, Methoden zu entwickeln, die die Quantisierung von Super- gravitationstheorien mit Hilfe von Techniken aus der Schleifenquantengravitation erm¨oglichen, um Fortschritte beim Vergleichen der Schleifenquantengravitation mit der Superstringtheorie zu machen. Die abstrakte Idee dieses Vergleichs ist es, sowohl die Schleifenquantengravitation als auch die Superstringtheorie als Quantisierungen einer zugrundeliegenden Gravitationstheorie zu verstehen. W¨ahrenddie Schleifenquantengravitation die ¨ublichen Probleme bei Fock-Typ Quan- tisierungen von Gravitationstheorien vermeidet, indem sie von einer anderen Poisson-*-Algebra von Phasenraumfunktionen startet, postuliert die Superstringtheorie den String als neuen fun- damentalen Baustein um den UV-Divergenzen in der perturbativen Quantengravitation Herr zu werden, welche durch die Betrachtungen von beliebig kleinen Distanzen entstehen. Das offensichtlichste Problem,
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