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