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Journal of Lie Theory 33 (2023), No. 2, 567--608
Copyright Heldermann Verlag 2023

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

Carlo Alberto Cremonini
Faculty of Mathematics and Physics, Mathematical Institute, Charles University Prague, Czech Republic

Pietro Antonio Grassi
(1) Dip. di Scienze e Innovazione Tecnologica, Università del Piemonte Orientale, Alessandria, Italy
(2) Arnold-Regge-Center, Torino, Italy, (3) INFN, Sezione di Torino, Torino, Italy

Simone Noja
(1) Mathematisches Institut, Universität Heidelberg, Germany
(2) Istituto Nazionale di Alta Matematica INdAM - GNSAGA, Roma, Italy

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 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-Eilenberg cohomology extended to integral forms and we prove that it is isomorphic via a Poincaré duality-type pairing to the ordinary Chevalley-Eilenberg cohomology of the Lie superalgebra. Next, we study equivariant Chevalley-Eilenberg 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.

Keywords: Lie superalgebras, Chevalley-Eilenberg cohomology, coset superspace, differential forms, integral forms.

MSC: 17B56, 17B81.

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