Generalized geometry of type B<sub>n</sub> is the study of geometric structures in T+T<sup>*</sup>+1, the sum of the tangent and cotangent bundles of a manifold and a trivial rank 1 bundle. The symmetries of this theory include, apart from B-fields, the novel A-fields. The relation between B<sub>n</sub>-geometry and usual generalized geometry is stated via generalized reduction. We show that it is possible to twist T+T<sup>*</sup>+1 by choosing a closed 2-form F and a 3-form H such that dH+F<sup>2</sup>=0. This motivates the definition of an odd exact Courant algebroid. When twisting, the differential on forms gets twisted by d+Fτ+H. We compute the cohomology of this differential, give some examples, and state its relation with T-duality when F is integral. We define B<sub>n</sub>-generalized complex structures (B<sub>n</sub>-gcs), which exist both in even and odd dimensional manifolds. We show that complex, symplectic, cosymplectic and normal almost contact structures are examples of B<sub>n</sub>-gcs. A B<sub>n</sub>-gcs is equivalent to a decomposition (T+T<sup>*</sup>+1)<sub>ℂ</sub>= L + L + U. We show that there is a differential operator on the exterior bundle of L+U, which turns L+U into a Lie algebroid by considering the derived bracket. We state and prove the Maurer-Cartan equation for a B<sub>n</sub>-gcs. We then work on surfaces. By the irreducibility of the spinor representations for signature (n+1,n), there is no distinction between even and odd B<sub>n</sub>-gcs, so the type change phenomenon already occurs on surfaces. We deal with normal forms and L+U-cohomology. We finish by defining G<sup>2</sup><sub>2</sub>-structures on 3-manifolds, a structure with no analogue in usual generalized geometry. We prove an analogue of the Moser argument and describe the cone of G<sup>2</sup><sub>2</sub>-structures in cohomology.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:669803 |
| Date | January 2014 |
| Creators | Rubio, Roberto |
| Contributors | Dancer, Andrew ; Hitchin, Nigel ; Gualtieri, Marco |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://ora.ox.ac.uk/objects/uuid:e0e48bb4-ea5c-4686-8b91-fcec432eb89a |
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