Generalised contact geometry as reduced generalised Complex geometry

Abstract

Generalised contact structures are studied from the point of view of reduced generalised complex structures, naturally incorporating non-coorientable structures as non-trivial fibering. The infinitesimal symmetries are described in detail, with a geometric description given in terms of gerbes. As an application of the reduction procedure, generalised coKähler structures are defined in a way which extends the Kähler/coKähler correspondence.1 where H ∈ Ω 3 cl (M ), is given by(for details see [5]). The Leibniz identity for • H (2.5a), gives the Maurer-Cartan identity dH = 0. The Dorfman product, • H , is natural in the sense that it is the derived bracket of d H := d + H∧ (with d the de Rham differential), acting on Γ(Equivalent exact Courant algebroids are classified by [H] ∈ H 3 (M, R), a point first noted by Severa [25]. The equivalence of the exact Courant algebroid under isotropic splittings corresponding to some B ∈ Ω 2 (M ) is closely related to the concepts of symmetries in generalised geometry structures. 2.1. Courant algebroid symmetries. Perhaps the most interesting aspect of generalised geometry is the enhanced symmetry group. The symmetry group of the Lie algebroid, given by the commutator of vector fields on T M , is Diff(M ). Exact Courant algebroids have a symmetry group given by Diff(M ) ⋉ Ω 2 cl (M ) if H = 0. Definition 2.3. A generalised almost complex structure on TM is given by J ∈ End(TM ), satisfying J * = −J and J 2 = −id.A generalised almost complex structure can equivalently be described by a maximal isotropic complex subbundle L ⊂ TM ⊗ C, satisfying L ∩L = {0}, forThere is a local one-to-one correspondence between generalised almost complex structures and conformal classes of complex pure spinors, where a complex pure spinor satisfies the non-degeneracy condition (φ, ϕ) M = 0.A generalised almost complex structure, J , is H-involutive if all sections X of the +i-eigenbundle L J are involutive with respect to • H : X 1 , X 2 ∈ L J ⇒ X 1 • H X 2 ∈ L J . Definition 2.4. A generalised complex structure is an H-involutive generalised almost complex structure.

Commuting Pairs of Generalized Contact Metric Structures

“…In light of our analysis above of the classical Sasakian case, we propose that a definition of a generalized Sasakian structure on a manifold M should be in terms of families of commuting pair of generalized almost contact metric structures (M, Φ, E ± , G) and (M, Φ, Ẽ± , G) satisfying the condition E ± = Ẽ± or the condition E ± = Ẽ∓ and such that Φ, and Φ are integrable with respect to a derived bracket. There have been previous definitions proposed for the notion of generalized Sasakian by Vaisman [16][17][18], Sekiya [15], Inglesias-Ponte and Wade [10], and Wright [19] all specific to the situation M × R + . Vasiman and Sekiya define Sasakian in terms of integrability of generalized almost complex structures on M × R + as opposed to defining the concept in terms of the intrinsic data on M .…”

Gauged sigma-models with nonclosed 3-form and twisted Jacobi structures

“…Here we will be interested in simpler situations and comment on the heterotic case in the concluding section. Nonclosed 3-forms may also arise upon dimensional reduction in the target space, for instance when it has the structure of a circle fibration [23,24] and more generally in reduction of Courant algebroids [25][26][27][28][29].…”

“…In the Abelian case, we find that gauge invariance of the gauged theory imposes additional constraints, as expected from the corresponding results with closed 3-form. These constraints acquire an elegant geometric interpretation when one considers the structure of a contact Courant algebroid over the extended vector bundle T M ⊕R⊕R⊕T * M twisted by three tensors, specifically a 3-form H and two 2-forms Ω and F whose product controls the 4-form dH [23,29]; they are identified with conditions for Dirac structures of the twisted contact Courant algebroid. The same geometric interpretation persists in the more general, non-Abelian case, with the difference that the corresponding higher structure is a nonexact Courant algebroid of the type described in [27], over an extended bundle T M ⊕ G ⊕ T * M , with G a bundle of quadratic Lie algebras.…”

B_n-generalized pseudo-Kahler structures

“…It is worth mentioning that there are various approaches to define odd dimensional analogues of D n -geometry (see e.g. [12,15,10]). In this paper we adopt the viewpoint of [11] and, in analogy with [8], we define generalized pseudo-Kähler structures on odd exact Courant algebroids as an enrichment of B ngeneralized complex structures.…”