Courant bracket twisted both by a 2-form B and by a bi-vector $$\theta $$

We obtain the Courant bracket twisted simultaneously by a 2-form B and a bi-vector $$\theta $$ θ by calculating the Poisson bracket algebra of the symmetry generator in the basis obtained acting with the relevant twisting matrix. It is the extension of the Courant bracket that contains well known Schouten–Nijenhuis and Koszul bracket, as well as some new star brackets. We give interpretation to the star brackets as projections on isotropic subspaces.

Courant bracket as T-dual invariant extension of Lie bracket

We consider the symmetries of a closed bosonic string, starting with the general coordinate transformations. Their generator takes vector components ξμ as its parameter and its Poisson bracket algebra gives rise to the Lie bracket of its parameters. We are going to extend this generator in order for it to be invariant upon self T-duality, i.e. T-duality realized in the same phase space. The new generator is a function of a 2D double symmetry parameter Λ, that is a direct sum of vector components ξμ, and 1-form components λμ. The Poisson bracket algebra of a new generator produces the Courant bracket in a same way that the algebra of the general coordinate transformations produces Lie bracket. In that sense, the Courant bracket is T-dual invariant extension of the Lie bracket. When the Kalb-Ramond field is introduced to the model, the generator governing both general coordinate and local gauge symmetries is constructed. It is no longer self T-dual and its algebra gives rise to the B-twisted Courant bracket, while in its self T-dual description, the relevant bracket becomes the θ-twisted Courant bracket. Next, we consider the T-duality and the symmetry parameters that depend on both the initial coordinates xμ and T-dual coordinates yμ. The generator of these transformations is defined as an inner product in a double space and its algebra gives rise to the C-bracket.

DEFORMATION QUANTIZATION OF POISSON MANIFOLDS IN THE DERIVATIVE EXPANSION

Deformation quantization of Poisson manifolds is studied within the framework of an expansion in powers of derivatives of Poisson structures. We construct the Lie group associated with a Poisson bracket algebra which defines a second order deformation in the derivative expansion.