We give a systematic derivation of the local expressions of the NS H-flux, geometric F -as well as non-geometric Q-and R-fluxes in terms of bivector β-and two-form Bpotentials including vielbeins. They are obtained using a supergeometric method on QP-manifolds by twist of the standard Courant algebroid on the generalized tangent space without flux. Bianchi identities of the fluxes are easily deduced. We extend the discussion to the case of the double space and present a formulation of T-duality in terms of canonical transformations between graded symplectic manifolds. Finally, the construction is compared to the formerly introduced Poisson Courant algebroid, a Courant algebroid on a Poisson manifold, as a model for R-flux.There are two dualities interrelating the various ten-dimensional superstring theories: Tduality and S-duality. Whereas S-duality relates strong and weak coupling regimes, T-duality exchanges winding and momentum modes of closed strings wrapping compact cycles and is a map between different string backgrounds. It is a target space symmetry.Additionally, fluxes wrapping the internal cycles of compactified string theories play an important role when considering T-duality. There is the NS-NS two-form B-field, to which the string couples, and its three-form field strength, the so-called H-flux. Furthermore, the f -flux is the torsion-less part of the projected spin-connection and therefore is closely related to the geometry of the compactified space itself.In the case, where the compactified space exhibits a Killing isometry, T-duality in this direction is possible and mixes B-field and metric components. The equations that express the new metric and B-field in terms of the old ones are the so-called Buscher rules [1, 2]. However, if one considers successive T-duality transformations of a three-torus with H-flux background, so-called non-geometric backgrounds with associated non-geometric fluxes Q and R appear [3, 4]. The Q-flux signalizes a globally non-geometric background with monodromy, which has to be patched by T-duality transformation. The R-flux signalizes an even locally non-geometric background, where standard manifold descriptions fail.Double field theory [5,6] approaches this problem by the introduction of winding coordinates, which are dual to the standard ones, and formulating T-duality on toroidal backgrounds as an O(D, D)-transformation on this doubled set of coordinates. In this formulation, even T-duality in non-isometry directions is possible and the non-geometric Q-and R-flux can be interpreted naturally [7,8]. The potential of R-flux is conjectured to be given by a bivector field β. A supergravity formulation making use of the β-potential can be found in [9].Since T-duality mixes metric and B-field, both structures can be combined in an O(D, D)tensor, the so-called generalized metric. It turns out that the associated backgrounds with H-flux can be naturally described using the Courant algebroid on the generalized tangent bundle T M ⊕ T * M. The underlying structure is given by g...

We construct a topological sigma model and a current algebra based on a Courant algebroid structure on a Poisson manifold. In order to construct models, we reformulate the Poisson Courant algebroid by supergeometric construction on a QP-manifold. A new duality of Courant algebroids which transforms H-flux and R-flux is proposed, where the transformation is interpreted as a canonical transformation of a graded symplectic manifold. Recently, there are further developments related to T-duality. Double field theory [4] is a manifestly O(d, d) covariant field theory which allows also for T-duality along non-isometry directions. Examples for other developments are the branes as sources for Q-and R-fluxes [5, 6] and the β-supergravity [7]. The topological T-duality [8, 9] is also proposed to analyze T-duality with flux. However, the background geometric structures for nongeometric fluxes are not well understood. A background geometry in string theory with NS H-flux [10] is known to be a Courant algebroid [11, 12], and the standard Courant algebroid of the generalized tangent bundle T M ⊕ T * M is of particular interest in the framework of generalized geometry [13, 14]. The T-duality on the H-flux is well understood as an automorphism on the standard Courant algebroid if ι X ι Y H = 0 [15]. However, we cannot simultaneously introduce all degrees of freedom of H-, F -, Q-, R-fluxes as deformation of the Courant algebroid. The only independent deformation in the exact Courant algebroid is a 3-form (H-flux) degree of freedom [16]. Recently, the Courant algebroid on a Poisson manifold, i.e. the Poisson Courant algebroid, has been introduced in [17] as a geometric object for a background with R-flux. It is shown that the nontrivial flux R of a 3-vector can be introduced consistently on a Poisson manifold as a deformation of the Courant algebroid. It is the 'contravariant object' [18] with respect to the standard Courant algebroid, which is the exchange of T * M with T M and H-flux with R-flux. The T-duality on the R-flux has also been analyzed and it has been shown that the duality of R-flux with Q-flux is also understood as an automorphism on the Poisson Courant algebroid [19].In this paper, we analyze the geometric structure of the Poisson Courant algebroid and a

We continue our exploration of local Double Field Theory (DFT) in terms of symplectic graded manifolds carrying compatible derivations and study the case of heterotic DFT. We start by developing in detail the differential graded manifold that captures heterotic Generalized Geometry which leads to new observations on the generalized metric and its twists. We then give a symplectic pre-NQ-manifold that captures the symmetries and the geometry of local heterotic DFT. We derive a weakened form of the section condition, which arises algebraically from consistency of the symmetry Lie 2-algebra and its action on extended tensors. We also give appropriate notions of twists -which are required for global formulations -and of the torsion and Riemann tensors. Finally, we show how the observed α ′ -corrections are interpreted naturally in our framework.

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