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 \(H\).