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Aspects of Generalized Geometry: Branes with Boundary, Blow-Ups, Brackets and Bundles

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Aspects of Generalized Geometry: Branes with Boundary, Blow-Ups, Brackets and Bundles Aspects of Generalized Geometry: Branes with Boundary, Blow-ups, Brackets and Bundles Charlotte Sophie Kirchhoff-Lukat Department of Applied Mathematics and Theoretical Physics University of Cambridge This dissertation is submitted for the degree of Doctor of Philosophy Trinity College September 2018 Declaration I hereby declare that except where specific reference is made to the work of others, the contents of this dissertation are original and have not been submitted in whole or in part for consideration for any other degree or qualification in this, or any other university. This dissertation is my own work and details on all collaborations are declared in Chapter 1. Charlotte Sophie Kirchhoff-Lukat September 2018 Acknowledgements I would like to extend sincere thanks to the many people who helped and supported me throughout my PhD research: My supervisor Professor Malcolm Perry showed great patience and encouraged me to find and pursue my own research interests. Professor Marco Gualtieri proposed the project that became Part I of this thesis, and supervised and guided me throughout my research. I further thank him and the Department of Mathematics at the University of Toronto for hosting me during a productive and intellectually stimulating long-term research visit. I learned mathematics at a faster rate during this time than at any other time in my mathematical education. Professor Madeleine Jotz-Lean advised me throughout the course of my PhD, and we had a productive and close collaboration which resulted in the publication on which Part II is based. Professor Gil Cavalcanti and the Mathematical Institute at Utrecht University hosted me during a short research visit. I value our discussions on toric and stable generalized complex Lefschetz fibrations. Dr. David Skinner, my departmental adviser at DAMTP, supported and guided me regarding the progression of my PhD, and gave helpful advice concerning my future career. Trinity College, Cambridge, has not only supported me financially with an Internal Grad- uate Studentship, but also provided an excellent academic and living environment. I further thank the college for awarding me the Rouse-Ball Travelling Scholarship in Mathematics, which enabled my research visit to the University of Toronto. Special thanks to Dr. Jean Khalfa and the whole tutorial team, who helped me both in academic and in personal matters. The STFC covered my university and college fees for the duration of my PhD degree, and the Cambridge Philosophical Society awarded me a Research Studentship to fund the continuation of my research in my final year. Professor Nigel Hitchin and Professor Ivan Smith each took the time to discuss my research project in-depth across several meetings, taught me about Hyperkähler geometry and the symplectic flux respectively, and advised me regarding the next steps. Sincere thanks to the examiners of my thesis, Dr. Ailsa Keating and Professor Gil Cavalcanti for their careful scrutiny of this text, for the mathematical discussion of the content, and for their helpful critique of my mathematical writing. Throughout my PhD I have had the opportunity to discuss various aspects of Poisson and symplectic geometry in particular, and geometry more broadly, with a number of inspiring mathematical colleagues. These discussions have helped to broaden my mathematical knowledge and interests. Dr. Ana Rita Pires, Professor Eva Miranda and Dr. Geoffrey Scott have helped me deepen my knowledge of log symplectic geometry, Francis Bischoff and Dr. Nikita Nikolaev answered many questions on various aspects of geometry, Davide Alboresi and I discussed pseudoholomorphic curves and Floer theory, and Alec Barns-Graham and I exchanged ideas on quantum field theory, flag manifolds, quivers and other areas where mathematics and physics intersect. Finally, I wish to thank my parents Christiane Kirchhoff and Rüdiger Lukat for their unwavering both practical and emotional support and encouragement in all matters during the entirety of my studies, my husband Hans Schmidt, and my close friends in the UK and in Germany, and in particular my fellow DAMTP PhD students Chandrima Ganguly and Markus Kunesch for making thesis-writing much more enjoyable and less solitary. vi Abstract This thesis explores aspects of generalized geometry, a geometric framework introduced by Hitchin and Gualtieri in the early 2000s. In the first part, we introduce a new class of submanifolds in stable generalized complex manifolds, so-called Lagrangian branes with boundary. We establish a correspondence between stable generalized complex geometry and log symplectic geometry, which allows us to prove results on local neighbourhoods and small deformations of this new type of submanifold. We further investigate Lefschetz thimbles in stable generalized complex Lefschetz fibrations and show that Lagrangian branes with boundary arise in this context. Stable generalized complex geometry provides the simplest examples of generalized complex manifolds which are neither complex nor symplectic, but it is sufficiently similar to symplectic geometry for a multitude of symplectic results to generalize. Our results on Lefschetz thimbles in stable generalized complex geometry indicate that Lagrangian branes with boundary are part of a potential generalisation of the Wrapped Fukaya category to stable generalized complex manifolds. The work presented in this thesis should be seen as a first step towards the extension of Floer theory techniques to stable generalized complex geometry, which we hope to develop in future work. The second part of this thesis studies Dorfman brackets, a generalisation of the Courant- Dorfman bracket, within the framework of double vector bundles. We prove that every Dorfman bracket can be viewed as a restriction of the Courant-Dorfman bracket on the standard VB-Courant algebroid, which is in this sense universal. Dorfman brackets have previously not been considered in this context, but the results presented here are reminiscent of similar results on Lie and Dull algebroids: All three structures seem to fit into a more general duality between subspaces of sections of the standard VB-Courant algebroid and brackets on vector bundles of the form TM ⊕ E∗,E → M a vector bundle. We establish a correspondence between certain properties of the brackets on one, and the subspaces on the other side. Contents Notation xiii 1 Introduction 1 2 Generalized Geometry 7 2.1 Courant algebroids ................................. 7 2.2 Leibniz algebroids ................................. 10 2.3 Generalized geometric structures ......................... 11 2.3.1 Generalized complex geometry ...................... 13 2.3.2 Generalized complex branes ........................ 17 2.4 Generalized geometry in theoretical high-energy physics ............ 18 2.4.1 Double and exceptional field theory ................... 19 2.4.2 String compactifications .......................... 21 3 Double vector bundles 23 3.1 Double vector bundles, duals and linear splittings ................ 23 3.2 The tangent double and the cotangent double of a vector bundle ....... 27 3.3 The first jet bundle of a vector bundle. ...................... 29 I Lagrangian branes and symplectic methods for stable generalized complex manifolds 31 4 Logarithmic and elliptic symplectic geometry 33 4.1 Logarithmic symplectic geometry ......................... 34 4.2 Complex log symplectic and elliptic symplectic geometry ............ 36 4.3 Stable generalized complex structures ...................... 41 4.3.1 Generalized complex branes in stable generalized complex manifolds . 42 5 Log and elliptic symplectomorphisms and flux 43 5.1 The groupoid exponential map .......................... 44 Contents 5.2 The Flux homomorphism for log and elliptic symplectic manifolds ...... 47 5.3 Lagrangian neighbourhood theorem for log symplectic manifolds ....... 52 5.4 Small deformations of Lagrangians in log symplectic manifolds ........ 55 5.4.1 Log and elliptic Lagrangians as coisotropic submanifolds in a Poisson manifold ................................... 57 6 Lagrangian branes with boundary in stable generalized complex manifolds 61 6.1 Definition and basic properties .......................... 61 6.2 Real oriented blow-up of the anticanonical divisor ................ 65 6.2.1 Lagrangian branes under blow-up ..................... 74 6.2.2 Neighbourhoods of Lagrangian branes with boundary ......... 77 6.3 Small deformations of Lagrangian branes with boundary ............ 80 7 Lefschetz thimbles in stable generalized complex Lefschetz fibrations 85 7.1 Ehresmann connections for log symplectic Lefschetz fibrations ......... 87 7.2 Stable generalized complex Lefschetz fibrations under blow-up ......... 90 7.3 Lefschetz thimbles in stable generalized complex Lefschetz fibrations and examples of Lagrangian branes with boundary .................. 92 7.3.1 Example: The Hopf surface ........................ 94 7.3.2 Examples of genus-one boundary Lefschetz fibrations over the disk .. 97 8 Lagrangian branes with boundary and complex branes in holomorphic log symplectic manifolds 101 8.1 Example: C × T 2 .................................. 102 8.2 Example: Hopf Surface ............................... 103 8.3 Example: CP 2 ................................... 106 9 Conclusions and Outlook: Stable Hamiltonian systems from stable gener- alized complex manifolds 109 II Dorfman brackets and double vector bundles 113 10 Brackets on TM ⊕ E∗ and the standard VB-Courant algebroid 115 10.1 Dorfman brackets and dull brackets ......................
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