Generalized geometry and the Hodge decomposition

In this lecture, delivered at the string theory and geometry workshop in Oberwolfach, we review some of the concepts of generalized geometry, as introduced by Hitchin and developed in the speaker’s thesis. We also prove a Hodge decomposition for the twisted cohomology of a compact generalized Kähler manifold, as well as a generalization of the dd-lemma of Kähler geometry. 1 Geometry of T ⊕ T ∗ The sum T ⊕T ∗ of the tangent and cotangent bundles of an n-dimensional manifold has a natural O(n, n) structure given by the inner product 〈X + ξ, Y + η〉 = 1 2 (ξ(Y ) + η(X)), and we may describe the Lie algebra in the usual way: so(n, n) = ∧T ⊕ End(T )⊕ ∧T . Hence we may view 2-forms B and bivectors β as infinitesimal symmetries of T ⊕ T . We may also form the Clifford algebra CL(T⊕T ), which has a spin representation on the Clifford module ∧T ∗ as described by Cartan: (X + ξ) · ρ = iXρ+ ξ ∧ ρ, for X+ ξ ∈ T ⊕T ∗ and ρ ∈ ∧T . This means that we may view differential forms as spinors for T ⊕T . From the general theory of spinors, this implies that there is a Spino(n, n)-invariant bilinear form 〈, 〉 : ∧ T ∗ × ∧T ∗ −→ det T , given by 〈α, β〉 = [α∧σ(β)]n , where σ is the anti-automorphism which reverses the order of wedge product. Another structure emerging from the interpretation of forms as spinors is the Courant bracket [, ]H on sections of T ⊕ T , obtained as the derived bracket (see [7]) of the natural differential operator d+H ∧ · acting on differential forms, where d is the exterior derivative and H ∈ Ωcl(M). When H = 0, we have the following: Proposition 1. The group of orthogonal automorphisms of the Courant bracket for H = 0 is a semidirect product of Diff(M) and Ωcl(M), where B ∈ Ω 2 cl(M) acts as the shear exp(B) on T ⊕ T . In this way we see that the natural appearance of H ∈ Ωcl(M) and the action of B ∈ Ω 2 cl(M) coincide precisely with the physicists’ description of the Neveu–Schwarz 3-form flux and the action of the B-field, respectively. Actually, the bundle of spinors differs from ∧•T ∗ by tensoring with the line bundle det T 1/2; we assume a trivialization has been chosen – this is related to the physicists’ dilaton field.