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

TL;DR

This paper explores the cohomology of Lie superalgebras relevant to supersymmetric theories, extending classical cohomology to integral forms and analyzing equivariant cohomology of coset superspaces with explicit examples.

Contribution

It introduces a generalized Chevalley-Eilenberg cohomology for integral forms and demonstrates its isomorphism to the classical cohomology, with applications to supergravity and superstring models.

Findings

Explicit cocycle expressions for Lie superalgebras

Generalization of cohomology to integral forms

Identification of infinite-dimensional cohomology in examples

Abstract

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