Geodesic
Cohomology of Lie Superalgebras: Forms, Integral Forms and Coset Superspaces
Carlo Alberto Cremonini1 ; Pietro Antonio Grassi2, 3 ; Simone Noja4, 5
1Faculty of Mathematics and Physics, Mathematical Institute, Charles University Prague, Czech Republic
2(1) Dip. di Scienze e Innovazione Tecnologica, Università del Piemonte Orientale, Alessandria, Italy
3(2) Arnold-Regge-Center, Torino, Italy, (3) INFN, Sezione di Torino, Torino, Italy
4(1) Mathematisches Institut, Universität Heidelberg, Germany
5(2) Istituto Nazionale di Alta Matematica INdAM - GNSAGA, Roma, Italy
Journal of Lie Theory, Tome 33 (2023) no. 2, pp. 567-608

Voir la notice de l'article provenant de la source Heldermann Verlag

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.
Classification : 17B56, 17B81
Mots-clés : Lie superalgebras, Chevalley-Eilenberg cohomology, coset superspace, differential forms, integral forms

Carlo Alberto Cremonini  1   ; Pietro Antonio Grassi  2 , 3   ; Simone Noja  4 , 5

1 Faculty of Mathematics and Physics, Mathematical Institute, Charles University Prague, Czech Republic
2 (1) Dip. di Scienze e Innovazione Tecnologica, Università del Piemonte Orientale, Alessandria, Italy
3 (2) Arnold-Regge-Center, Torino, Italy, (3) INFN, Sezione di Torino, Torino, Italy
4 (1) Mathematisches Institut, Universität Heidelberg, Germany
5 (2) Istituto Nazionale di Alta Matematica INdAM - GNSAGA, Roma, Italy 
Carlo Alberto Cremonini; Pietro Antonio Grassi; Simone Noja. Cohomology of Lie Superalgebras: Forms, Integral Forms and Coset Superspaces. Journal of Lie Theory, Tome 33 (2023) no. 2, pp. 567-608. http://geodesic.mathdoc.fr/item/JOLT_2023_33_2_a5/
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     author = {Carlo Alberto Cremonini and Pietro Antonio Grassi and Simone Noja},
     title = {Cohomology of {Lie} {Superalgebras:} {Forms,} {Integral} {Forms} and {Coset} {Superspaces}},
     journal = {Journal of Lie Theory},
     pages = {567--608},
     year = {2023},
     volume = {33},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JOLT_2023_33_2_a5/}
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