Advances in Theoretical and Mathematical Physics

Volume 27 (2023)

Number 3

Generalised cocycles and super $p$-branes

Pages: 829 – 879

DOI: https://dx.doi.org/10.4310/ATMP.2023.v27.n3.a6

Authors

C. A. Cremonini (Faculty of Mathematics and Physics, Mathematical Institute, Charles University Prague, Czech Republic)

P. A. Grassi (Dipartimento di Scienze e Innovazione Tecnologica (DiSIT), Università del Piemonte Orientale, Alessandria, Italy; INFN, Sezione di Torino, Italy; and Arnold Regge Center, Torino, Italy)

Abstract

We study the cohomology (cocycles) of Lie superalgebras for the generalised complex of forms: superforms, pseudoforms and integral forms.We argue that these cocycles might be interpreted in the light of a new brane scan as generators of new higher-WZW terms and might provide new sources for supergravity. We use the technique of spectral sequences to abstractly compute the Chevalley–Eilenberg cohomology. We first focus on the superalgebra $\mathfrak{osp}(2|2)$ and show that there exist non-empty cohomology spaces among pseudoforms related to sub-superalgebras. We then extend some classical theorems by Koszul to include pseudoforms and integral forms. Further, we conjecture that the Poincaré duality extends to Lie superalgebras, as long as all the complexes of forms are taken into account and we prove that this holds for $\mathfrak{osp}(2|2)$. We finally construct the cohomology representatives explicitly by using a distributional realisation of pseudoforms and integral forms. On one hand, these results show that the cohomology of Lie superalgebras is larger than expected; on the other hand, we show the emergence of completely new cohomology classes represented by pseudoforms. These classes represent integral form classes of sub-superstructures.

Published 6 June 2024