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

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