Aspects of Generalized Geometry: Branes with Boundary, Blow-ups, Brackets and Bundles
Perry, Malcolm J.
University of Cambridge
Doctor of Philosophy (PhD)
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Kirchhoff-Lukat, C. S. (2018). Aspects of Generalized Geometry: Branes with Boundary, Blow-ups, Brackets and Bundles (Doctoral thesis). https://doi.org/10.17863/CAM.30372
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 T M ⊕ 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.
differential geometry, generalized complex geometry, Poisson geometry, symplectic geometry
Supported by an STFC Postgraduate Studentship and an Internal Graduate Studentship from Trinity College, Cambridge.
This record's DOI: https://doi.org/10.17863/CAM.30372
All rights reserved, Part I of this thesis is based on joint work with Professor Marco Gualtieri as detailed in Chapter 1. Part II of this thesis, as well as Chapter 2 and 3 contain extracts from the pre-print "Natural lifts of Dorfman brackets" available online at arXiv:1610.05986, jointly written with Professor Madeleine Jotz-Lean. A publication agreement is attached.