Starting with minimal requirements from the physical experience with higher gauge theories, i.e. gauge theories for a tower of differential forms of different form degrees, we discover that all the structural identities governing such theories can be concisely recombined into a so-called Q-structure or, equivalently, an L ∞ -algebroid. This has many technical and conceptual advantages: Complicated higher bundles become just bundles in the category of Q-manifolds in this approach (the many structural identities being encoded in the one operator Q squaring to zero), gauge transformations are generated by internal vertical automorphisms in these bundles and even for a relatively intricate field content the gauge algebra can be determined in some lines only and is given by the so-called derived bracket construction.This article aims equally at mathematicians and theoretical physicists; each more physical section is followed by a purely mathematical one. While the considerations are valid for arbitrary highest form-degree p, we pay particular attention to p = 2, i.e. 1-and 2-form gauge fields coupled non-linearly to scalar fields (0-form fields). The structural identities of the coupled system correspond to a Lie 2-algebroid in this case and we provide different axiomatic descriptions of those, inspired by the application, including e.g. one as a particular kind of a vector-bundle twisted Courant algebroid.

a b s t r a c tWe define a new kind of algebroid which fulfills a Leibniz rule, a Jacobi identity twisted by a 3-form H with values in the kernel of the anchor map, and the twist is closed under a naturally occurring exterior covariant derivative. We give examples and define three kinds of cohomologies, two via realizations as Q -structures on graded manifolds. The paper classifies PQ -manifolds of maximal degree 3.

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 ...

No abstract

Abstract. We introduce the notion of matched pairs of Courant algebroids and give several examples arising naturally from complex manifolds, holomorphic Courant algebroids, and certain regular Courant algebroids. We consider the matched sum of two Dirac subbundles, one in each of two Courant algebroids forming a matched pair.