Courant bracket found out to be T-dual to Roytenberg bracket

Bosonic string moving in coordinate dependent background fields is considered. We calculate the generalized currents Poisson bracket algebra and find that it gives rise to the Courant bracket, twisted by a 2-form $$2B_{\mu \nu }$$. Furthermore, we consider the T-dual generalized currents and obtain their Poisson bracket algebra. It gives rise to the Roytenberg bracket, equivalent to the Courant bracket twisted by a bi-vector $$\Pi ^{\mu \nu }$$, in case of $$\Pi ^{\mu \nu } = 2 {^\star B}^{\mu \nu } = \kappa \theta ^{\mu \nu }$$. We conclude that the twisted Courant and Roytenberg brackets are T-dual, when the quantities used for their deformations are mutually T-dual