Cohomology of Lie Superalgebras: Forms, Integral Forms and Coset Superspaces

We study Chevalley-Eilenb erg cohomology of physically relevant Lie superalgebras related to supersymmetric theories, providing explicit expressions for their cocycles in terms of their Maurer-Cartan forms. We include integral forms in the picture by defining the notions of constant densities and integral forms related to a Lie superalgebra. We develop a suitable generalization of Chevalley-Eilenb erg cohomology extended to integral forms and we prove that it is isomorphic via a Poincare duality-type pairing to the ordinary Chevalley-Eilenb erg cohomology of the Lie superalgebra. Next, we study equivariant Chevalley-Eilenb erg cohomology for coset superspaces, which play a crucial role in supergravity and superstring models. Again, we treat explicitly several examples, providing cocycles' expressions and revealing a characteristic infinite -dimensional cohomology.