The twisted gauge-natural bilinear brackets on couples of linear vector fields and linear p-forms

Abstract

We completely describe all gauge-natural operators $C$ which send linear $(p+2)$-forms $H$ on vector bundles $E$ (with sufficiently large dimensional bases) into $\mathbf{R}$-bilinear operators $C_H$ transforming pairs $(X_1\oplus\omega_1,X_2\oplus\omega_2)$ of couples of linear vector fields and linear $p$-forms on $E$ into couples $C_H(X_1\oplus\omega_1, X_2\oplus\omega_2)$ of linear vector fields and linear $p$-forms on $E$. Further, we extract all $C$ (as above) such that $C_0$ is the restriction of the well-known Courant bracket and $C_H$ satisfies the Jacobi identity in Leibniz form for all closed linear $(p+2)$-forms $H$.