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Generalised Geometry of Supergravity

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Generalised Geometry of Supergravity Imperial College London Department of Physics Generalised Geometry of Supergravity Charles Strickland-Constable August 1, 2012 Supervised by Professor Daniel Waldram Submitted in part fulfilment of the requirements for the degree of Doctor of Philosophy in Physics of Imperial College London and the Diploma of Imperial College London 1 2 Declaration I herewith certify that, to the best of my knowledge, all of the material in this dissertation which is not my own work has been properly acknowledged. The research described has been done in collaboration with Daniel Waldram and Andr´eCoimbra, and the presentation in chapters 3-6 follows closely the papers [1, 2, 3]. Charles Strickland-Constable 3 4 Abstract We reformulate type II supergravity and dimensional restrictions of eleven- dimensional supergravity as generalised geometrical analogues of Einstein gravity. The bosonic symmetries are generated by generalised vectors, while the bosonic fields are unified into a generalised metric. The generalised tan- gent space features a natural action of the relevant (continuous) duality group. Also, the analogues of orthonormal frames for the generalised met- ric are related by the well-known enhanced local symmetry groups, which provide the analogue of the local Lorentz symmetry in general relativity. Generalised connections and torsion feature prominently in the construc- tion, and we show that the analogue of the Levi-Civita connection is not uniquely determined by metric compatibility and vanishing torsion. How- ever, connections of this type can be used to extract the derivative operators which appear in the supergravity equations, and the undetermined pieces of the connection cancel out from these, leaving the required unique expres- sions. We find that the bosonic action and equations of motion can be inter- preted as generalised curvatures, while the derivative operators appearing in the supersymmetry variations and equations of motion for the fermions become very simple expressions in terms of the generalised connection. In the final chapter, the construction is used to reformulate supersym- metric flux backgrounds as torsion-free generalised G-structures. This is the direct analogue of the special holonomy condition which arises for su- persymmetric backgrounds without flux in ordinary Riemannian geometry. 5 In loving memory of Polly Strickland-Constable 6 Acknowledgements There are many people who have been of great help and support to me during my studies. First and foremost, I would like to thank my supervisor, Daniel Wal- dram, for many enlightening meetings and great patience. It seems rare to encounter someone who has both amazing insight and the ability to share their ideas so coherently, and I feel extremely lucky to have had such a great mentor. I would also like to thank my colleagues and friends at Imperial. You have made Imperial a great place to be during my time here. On the academic front, I should express special thanks to Andr´eCoimbra, with whom I have spent many enjoyable hours discussing the material in this thesis (as well as football and many other things). I have also benefitted greatly from collaboration and discussions with Tom Pugh, concerning many topics in theoretical physics. Thanks also to Kelly Stelle for collaboration and discussions, and to Daniel Thomas for being a great office-mate. Finally, I cannot express enough gratitude to my parents Fred and Polly, whose unwavering support has always gone beyond the extra mile and made everything possible for me. 7 Contents 1. Introduction 11 2. Differential Geometry and Gravity 32 2.1. Metric structures, torsion and the Levi{Civita connection . 32 2.2. General Relativity . 35 2.3. NS-NS Sector of Type II Supergravity . 37 + 3. O(d; d) × R Generalised Geometry 40 3.1. T ⊕ T ∗ Generalised Geometry . 40 3.1.1. Linear algebra of F ⊕ F ∗ and SO(d; d)......... 40 3.1.2. Differential structures on T ⊕ T ∗ ............ 42 + 3.2. O(d; d) × R Generalised Geometry . 44 3.2.1. Generalised structure bundle . 44 3.2.2. Generalised tensors and spinors and split frames . 46 3.2.3. The Dorfman derivative, Courant bracket and exte- rior derivative . 49 + 3.2.4. Generalised O(d; d) × R connections and torsion . 51 3.3. O(p; q) × O(q; p) structures and torsion-free connections . 55 3.3.1. O(p; q) × O(q; p) structures and the generalised metric 55 3.3.2. Torsion-free, compatible connections . 58 3.3.3. Unique operators and generalised O(p; q) × O(q; p) curvatures . 61 + 4. Type II Theories as O(10; 10) × R Generalised Geometry 64 4.1. Type II supergravity . 64 4.2. Type II supergravity as O(9; 1) × O(1; 9) generalised gravity . 68 4.2.1. NS-NS and fermionic supergravity fields . 68 4.2.2. RR fields . 70 4.2.3. Supersymmetry variations . 72 8 4.2.4. Equations of motion . 73 + 5. Ed(d) × R Generalised Geometry 76 + 5.1. Ed(d) ×R generalised tangent space . 76 5.1.1. Generalised bundles and frames . 77 5.1.2. The Dorfman derivative and Courant bracket . 83 + 5.1.3. Generalised Ed(d) ×R connections and torsion . 85 5.1.4. The \section condition", Jacobi identity and the ab- sence of generalised curvature . 87 5.2. Hd structures and torsion-free connections . 89 5.2.1. Hd structures and the generalised metric . 89 5.2.2. Torsion-free, compatible connections . 91 5.2.3. Unique operators and generalised Hd curvatures . 93 + 6. Dimensional restrictions of D = 11 Supergravity as Ed(d) ×R Generalised Geometry 97 6.1. Dimensional restrictions of eleven-dimensional supergravity . 97 6.1.1. Eleven dimensional supergravity . 97 6.1.2. Restriction to d dimensions . 98 6.1.3. Action, equations of motion and supersymmetry . 99 6.2. Supergravity degrees of freedom and Hd structures . 102 6.3. Supergravity equations from generalised geometry . 105 6.4. Realisation in SO(d) representations . 106 6.5. Comments on type II theories . 111 7. Supersymmetric backgrounds as generalised G-structures 113 7.1. Holonomy, G-structures and Intrinsic Torsion . 113 7.2. Supersymmetric Backgrounds of String Theory and M Theory 117 7.2.1. 4D Minkowski compactifications of type II theories . 119 7.2.2. 4D warped Minkowski solutions of M-theory . 124 7.3. Generalised Complex Geometry and Supersymmetric Vacua . 126 7.4. Killing Spinor Equations and Generalised Holonomy . 128 7.4.1. SU (3) × SU (3) structures in type II theories . 128 7.4.2. SU (7) and SU (6) structures in M-theory . 130 7.4.3. Computation of the Intrinsic Torsion . 131 8. Conclusion and Outlook 136 9 A. General Conventions 141 B. Clifford Algebras 143 + C. Details of Ed(d) ×R and Hd 146 + C.1. Ed(d) ×R and GL(d; R)..................... 146 + C.1.1. Construction of Ed(d) ×R from GL(d; R)....... 146 C.1.2. Some tensor products . 148 C.2. Hd and O(d)........................... 150 C.2.1. Construction of Hd from SO(d)............. 150 ~ C.2.2. Hd and Cliff(10; 1; R).................. 152 D. Group Theory Proof of Uniqueness of Operators 155 E. Spinor Decompositions 163 Bibliography 165 10 1. Introduction In this thesis, we will present generalised geometrical descriptions of su- pergravity theories which arise in the study of superstring theory at low energy. These new formulations write supergravity with the same geomet- ric structure as Einstein's theory of gravity and unify the bosonic degrees of freedom. Simultaneously, the hidden symmetries of supergravity appear in the construction. In this introductory chapter, we outline the historical de- velopment and motivations behind this research, and also discuss significant precursors and related works. Quantum mechanics and gravity The central problem of modern theoretical physics is to find a unified quan- tum theory which describes all observed physical interactions. The stan- dard model of particle physics describes the microscopic quantum mechan- ical behaviour of the elementary particles seen in accelerator experiments to a staggering degree of precision. At its core are the quantisations of some particular gauge field theories in a fixed special relativistic spacetime background. However, it does not contain gravity, whose quantisation is non-renormalisable when viewed as a conventional field theory. Currently, our best experimentally verified theory of gravity is Einstein's theory of general relativity, which explains how gravitational effects are due to the curvature of spacetime. This curvature is determined, via Einstein's field equation, from the configuration of matter and energy it contains, so spacetime itself becomes a dynamical element, in contrast to its role as a fixed background in the standard model. The equations governing this dy- namical spacetime have a geometrical structure, which makes the theory particularly elegant. However, this is a purely classical deterministic the- ory and the scales on which gravitational effects are observable reflects this. Gravity is a very weak force compared to the others in the standard model, and there is a natural scale at which quantum gravity effects are expected 11 to become important. This scale is constructed from the fundamental con- stants h (Planck's constant), c (speed of light) and G (Newton's constant) which characterise the theory. It can be expressed as the Planck length (∼ 10−33cm) or the Planck mass (∼ 1019GeV). As these scales are totally inaccessible to instruments made with existing technology, it is currently impossible to probe the quantum nature of gravity directly in experiments. However, there are situations in which such quantum effects would play an important role, thus a quantum theory is a requirement on more than just theoretical grounds. One is the microscopic description of quantum properties of black holes. A more fundamental problem is how to describe spacetime correctly in the very early universe and resolve the physics of the big-bang singularity predicted by general relativity.
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