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It is known that the topological T-duality exchanges $H$ and $F$-fluxes. In this paper, we reformulate the topological T-duality as an exchange of two Lie algebroids in the generalized tangent bundle. Then, we apply the same formulation to the Poisson-generalized geometry, which is introduced in arXiv:1408.2649 to define $R$-fluxes as field strength associated with $\beta$-transformations. We propose a definition of $Q$-flux associated with $\beta$-diffeomorphisms, and show that the topological T-duality exchanges $R$ and $Q$-fluxes., Comment: 30 pages

We study a new kind of Courant algebroid on Poisson manifolds, which is a variant of the generalized tangent bundle in the sense that the roles of tangent and the cotangent bundle are exchanged. Its symmetry is a semidirect product of $\beta$-diffeomorphisms and $\beta$-transformations. It is a starting point of an alternative version of the generalized geometry based on the cotangent bundle, such as Dirac structures and generalized Riemannian structures. In particular, $R$-fluxes are formulated as a twisting of this Courant algebroid by a local $\beta$-transformations, in the same way as $H$-fluxes are the twist of the generalized tangent bundle. It is a $3$-vector classified by Poisson $3$-cohomology and it appears in a twisted bracket and in an exact sequence., Comment: 22 pages

The properties of the D-brane fluctuations are investigated using the two types of deformation of the Dirac structure, based on the B-transformation and the beta-transformation, respectively. The former gives the standard gauge theory with 2-form field strength. The latter gives a non-standard gauge theory on the Poisson manifold with bivector field strength and the vector field as a gauge potential, where the gauge symmetry is a diffeomorphism generated by the Hamiltonian vector field. The map between the two gauge theories is also constructed with the help of Moser's Lemma and the Magnus expansion. We also investigate the relation to the gauge theory on the noncommutative D-branes., Comment: 39 pages, 5 figures