Title:

Generalized geometry of type Bn

Generalized geometry of type B_{n} 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 Bfields, the novel Afields. The relation between B_{n}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 2form F and a 3form H such that dH+F^{2}=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 Tduality when F is integral. We define B_{n}generalized complex structures (B_{n}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_{n}gcs. A B_{n}gcs is equivalent to a decomposition (T+T^{*}+1)_{ℂ}= 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 MaurerCartan equation for a B_{n}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_{n}gcs, so the type change phenomenon already occurs on surfaces. We deal with normal forms and L+Ucohomology. We finish by defining G^{2}_{2}structures on 3manifolds, a structure with no analogue in usual generalized geometry. We prove an analogue of the Moser argument and describe the cone of G^{2}_{2}structures in cohomology.
