Cohomology of Courant algebroids with split base

Abstract

In this paper we study the cohomology H • st (E) of a Courant algebroid E. We prove that if E is regular, H • st (E) coincides with the naive cohomology H • naive (E) of E as conjectured by Stiénon and Xu [SX08]. For general Courant algebroids E we define a spectral sequence converging to H • st (E). If E is with split base, we prove that there exists a natural transgression homomorphism T 3 (with image in H 3 naive (E)) which, together with H • naive (E), gives all H • st (E). For generalized exact Courant algebroids, we give an explicit formula for T 3 depending only on theŠevera characteristic clas of E. 1 right framework for pseudo-metric vector bundles equipped with something like an ad-invariant Lie algebra structure. A primary example of a Courant algebroid is given by the double A ⊕ A * of a Lie bialgebroid A or a Lie quasi-bialgebroid, that is a Lie bialgebroid twisted by a 3-form [Roy99]. An important step forward by Roytenberg was a description of Courant algebroids in terms of a derived bracket as introduced by Kosmann-Schwarzbach in [KS96], see [Roy01], or equivalently in terms of a nilpotent odd operator (also known as Q-structure). Hence the Courant algebroid structure with its intricate axioms can all be encoded in a cubic function H on a graded symplectic manifold and its derived bracket. To do so, one goes into the context of graded manifolds and considers the graded manifold E[1]. The pseudo-metric on E makes E[1] into a (graded) Poisson manifold. By constructions of Weinstein,Ševera [Šev] and Roytenberg [Roy01], there is a minimal symplectic realization (E, {., .}) of E[1]. Now, there is a cubic Hamiltonian H satisfying {H, H} = 0 on E encoding the Courant algebroid structure together with the symplectic structure of E. For instance, the Courant bracket is given by the formula [φ, ψ] = {{H, φ}, ψ}.The derived bracket construction also leads to a natural notion of cohomology of a Courant algebroid. Since {H, H} = 0, the operator Q = {H, .} : C ∞ (E) → C ∞ (E) is a differential. Hence one can define the cohomology of E as the cohomology H • (C ∞ (E), Q). So far, there are only few examples of Courant algebroids for which the cohomology is known. For instance when E ∼ = T * M ⊕ T M is an exact Courant algebroid, its cohomology is isomorphic to the de Rham cohomology of M . On the other hand, if the base is a point, E is a Lie algebra (together with an ad-invariant pseudo-metric) and its cohomology is isomorphic to its cohomology as a Lie algebra. One reason which makes H • (C ∞ (E), Q) rather difficult to treat is that its construction relies on the minimal symplectic realization E and not just on E or E * itself. In particular it is quite different from the usual cohomology theories for "Lie theoretic objects" such as Lie algebroids or Leibniz algebras where the cohomology is defined using a differential given by a Cartan type formula. For instance, the (de Rham) cohomology of a Lie algebroid (A, [., .], ρ : A → M ) over M is the cohomology of the complex of forms (Γ(Λ • A * ), d A ), where the ...

Courant–Dorfman Algebras and their Cohomology

“…This was proved by Ginot and Grutzmann [10] who also obtained further results by considering the spectral sequence associated to the filtration of C(M(E)) by the powers of what they called "the naive ideal". This ideal corresponds under to the ideal I we defined in Section 4.5.…”

“…In particular, we describe natural filtrations and subcomplexes, related to those considered in [10] and [24], which we expect to be an important tool in cohomology computations; derive commutation relations among certain derivations of C(E, R), similar to the well-known Cartan relations among contractions and Lie derivatives by vector fields; classify central extensions of the CourantDorfman algebra E in terms of H 2 (E, R). We also consider the canonical cocycle 1 (e; f ) = −ρ(e) f generalizing the Cartan 3-form on a quadratic Lie algebra appearing in the ChernSimons theory.…”

The standard cohomology of regular Courant algebroids

“…As an early attempt, Stiénon and Xu defined in [48] the naive complex (C • naive (E), d) of a Courant algebroid E, and they proved that the corresponding degree 1 naive cohomology H 1 naive (E) is isomorphic to the standard cohomology H 1 st (E). Later, Ginot and Grützmann proved in [19] that the naive cohomology of any transitive Courant algebroid is isomorphic to its standard cohomology, which was first conjectured by Stiénon and Xu in op.cit.. However, for general Courant algebroids, the two cohomology theories are quite different.…”

“…However, for general Courant algebroids, the two cohomology theories are quite different. Another achievement of [19] is the construction of a Leray type spectral sequence, called the naive ideal spectral sequence, to calculate the standard cohomology H • st (E) under the assumption that E has a split base.…”

On regular Courant algebroids

“…To this day, little is known about Courant algebroid cohomology [13]: Roytenberg computed it for T M ⊕ T * M and Ginot and Grützmann for transitive and some very special regular Courant algebroids [7]. Our result should be useful for computing the Courant algebroid cohomology of arbitrary regular Courant algebroids.…”