Abelian Extensions of the Group of Diffeomorphisms of a Torus

In this Letter we construct Abelian extensions of the group of diffecomorphisms of a torus. We consider the Jacobian map, which is a crossed homomorphism from the group of diffeomorphisms into a toroidal gauge group. A pull-back under this map of an invariant central 2-cocycle on a gauge group turns out to be an Abelian cocycle on the group of diffeomorphisms. In the case of a circle we get an interpretation of the Virasoro–Bott cocycle as a pull-back of the Heisenberg cocycle. We also give an Abelian generalization of the Virasoro–Bott cocycle to the case of a manifold with a volume form.