  In this paper we construct abelian extensions of the group of diffeomorphisms
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 a central 2-cocycle on a gauge group turns out to be an abelian
cocycle on the group of diffeomorphisms. We show that in the case of a circle,
the Virasoro-Bott cocycle is 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.

Group Theory