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Title: Generalized geometry of type Bn
Author: Rubio, Roberto
ISNI:       0000 0004 5369 5432
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2014
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Generalized geometry of type Bn is the study of geometric structures in T+T*+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 Bn-geometry and usual generalized geometry is stated via generalized reduction. We show that it is possible to twist T+T*+1 by choosing a closed 2-form F and a 3-form H such that dH+F2=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 Bn-generalized complex structures (Bn-gcs), which exist both in even and odd dimensional manifolds. We show that complex, symplectic, cosymplectic and normal almost contact structures are examples of Bn-gcs. A Bn-gcs is equivalent to a decomposition (T+T*+1)= 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 Bn-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 Bn-gcs, so the type change phenomenon already occurs on surfaces. We deal with normal forms and L+U-cohomology. We finish by defining G22-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 G22-structures in cohomology.
Supervisor: Dancer, Andrew ; Hitchin, Nigel ; Gualtieri, Marco Sponsor: Fellowship for Graduate Courses in Universities and Colleges of Fundación Caja Madrid
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Mathematics ; 3-manifold ; almost contact geometry ; complex geometry ; deformation theory ; G2(2)-structure ; generalized complex geometry ; twisted cohomology ; generalized geometry ; Lie algebroid