Generalized geometry of type Bn

Thesis

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 B-fields, the novel A-fields. 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 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 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 + L̄ + 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_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+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.

Authors: Rubio, R

• DOI: 10.5287/ora-2z7yba5jr
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: deformation theory; complex geometry; generalized geometry; almost contact geometry; generalized complex geometry; twisted cohomology; Lie algebroid; 3-manifold; G2(2)-structure
• Subjects: Mathematics