On the third cohomology of the Lie algebra of vector fields on weighted densities on R

Let Vect (R) be the Lie algebra of smooth vector fields on R and F_ be the space of -densities on R. Vect (R) acts on F_ by Lie derivative. In this paper, we compute the third differential cohomology of the Lie algebra Vect (R) with coeffcients in the space F_. Explicit cocycles spanning these cohomology spaces are given.