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

TL;DR

This paper derives a generalized Courant bracket twisted by both a 2-form and a bi-vector, extending known brackets and introducing new star brackets with geometric interpretations.

Contribution

It introduces a novel combined twisting of the Courant bracket by a 2-form and a bi-vector, unifying and extending existing bracket structures.

Findings

Derived the twisted Courant bracket with both B-field and bi-vector

Identified new star brackets as projections on isotropic subspaces

Connected the brackets to known structures like Schouten-Nijenhuis and Koszul

Abstract

We obtain the Courant bracket twisted simultaneously by a 2-form $B$ and a bi-vector $\theta$ 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.