AV-Courant Algebroids and Generalized CR Structures

Abstract

Abstract. We construct a generalization of Courant algebroids that are classified by the third cohomology group H 3 (A, V ), where A is a Lie Algebroid, and V is an A-module. We see that both Courant algebroids and E 1 (M) structures are examples of them. Finally we introduce generalized CR structures on a manifold, which are a generalization of generalized complex structures, and show that every CR structure and contact structure is an example of a generalized CR structure.

Conformal Courant Algebroids and Orientifold T-Duality

“…This allows us to work with manifolds rather than orbifolds, but we fully expect that our results can be extended to the general orientifold setting. [15] and the AV -Courant algebroids of [9]. We classify exact conformal Courant algebroids in Proposition 2.8.…”

Lie bialgebroids of generalized CRF-manifolds

“…We will explain their relationship with generalized contact structures. We refer the reader to [2] for more details about AV -Courant algebroids and generalized CR-structures.…”

“…Then A = T M ⊕ T * M is an AV -Courant algebroid[2].It is known that A is an exact Courant algebroid and we have:0 → V ⊗ T * M j → A → T M → 0, where V ⊗ T * M T * M and j ≡ * .Take H := ker η, identify H * with ker F , and set H = ker η ⊕ ker F . Then the endomorphism J : H → H whose components are ϕ, π , θ , and −ϕ * determines a generalized CR-structure in Li-Bland's sense[2] if the following two conditions hold:(i) J + J * = 0 and J 2 = −I; (ii) The space of sections of the subbundle K := q −1 (ker(J − iI)) is closed under [[−,−]], where q : −1 (H) → −1 (H)/ j(V ⊗ Ann(H)) is the natural projection. Since K := q −1 (ker(J − iI)) corresponds to L * := L η ⊕ E (1,0) , there follows: Remark.…”

Variation of Hodge structure for generalized complex manifolds

“…It is straightforward to see that this structure makes E into a family of exact Courant algebroids. From [1] we know that every family of exact Courant algebroids is isomorphic to one of this form for some unique class h ∈ H 3 (V, R). In analogy with the case of exact Courant algebroids we call the class h ∈ H 3 (V, R) theŠevera class of E. In general the map H 3 (M, R) → H 3 (V, R) is neither injective nor surjective so we would like to clarify the relation between these two groups.…”

“…Note however that when considering families it is better to think of an arbitrary generalized complex structure of complex type to look like a B-shift of (5.1), since we can always deform such a J by a family of B-shifts. Let H denote the closed 3-form corresponding to the choice of splitting for E. Then integrability of J is equivalent to integrability of I together with the restriction that H be of type (2, 1) + (1,2). The i eigenspace L is given by L = T 0,1 M ⊕ T * 1,0 M .…”