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A review of Lie superalgebra cohomology for pseudoforms. (English) Zbl 07655748

Summary: This note is based on a short talk presented at the “42nd Winter School Geometry and Physics” held in Srni, Czech Republic, January 15th–22nd 2022. We review the notion of Lie superalgebra cohomology and extend it to different form complexes, typical of the superalgebraic setting. In particular, we introduce pseudoforms as infinite-dimensional modules related to sub-superalgebras. We then show how to extend the Koszul-Hochschild-Serre spectral sequence for pseudoforms as a computational method to determine the cohomology groups induced by sub-superalgebras. In particular, we show as an example the case of \(\mathfrak{osp}(1\mid 4)\) and choose \(\mathfrak{osp}(1\mid 2)\times\mathfrak{sp}(2)\) as sub-algebra. We finally comment on some physical applications of such new cohomology classes related to super-branes. The note is a compact version of [C. A. Cremonini and P. A. Grassi, “Generalised cocycles and super \(p\)-branes”, arXiv:2206.03394 [hep-th]].

MSC:

17B56 Cohomology of Lie (super)algebras
17B81 Applications of Lie (super)algebras to physics, etc.
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