Fluxes of Courant Bracket Twisted at the Same Time by B $B$ and θ $\theta$

Abstract

The simultaneous twisting of the Courant bracket by a 2-form $B$ and a bi-vector $\theta$ is investigated, with the generalized fluxes obtained in Courant algebroid relations explored. The twisted Lie bracket is defined and it is demonstrate that the generalized $H$-flux can be expressed as the field strength defined on this Lie algebroid. Similarly, it is shown that the $f$-flux is a direct consequence of simultaneous twisting, which arises in the twisted Lie bracket relations between holonomic partial derivatives. The generalized $Q$ flux is obtained in terms of the twisted Koszul bracket, which is identified as a quasi-Lie algebroid bracket. The action of an exterior derivative related to the twisted Koszul bracket on a bi-vector produces the generalized $R$-flux. It is shown that the generalized $R$-flux is also the twisted Schouten–Nijenhuis bracket, i.e., the natural graded bracket on multi-vectors defined with the twisted Lie bracket.