Fermionic Spencer Cohomologies of D=11 Supergravity
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| Format: | Preprint |
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2024
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| author | Cremonini, C. A. Grassi, P. A. Noris, R. Ravera, L. Santi, A. |
| author_facet | Cremonini, C. A. Grassi, P. A. Noris, R. Ravera, L. Santi, A. |
| contents | We combine the theory of Cartan-Tanaka prolongations with the Molien-Weyl integral formula and Hilbert-Poincaré series to compute the Spencer cohomology groups of the $D=11$ Poincaré superalgebra $\mathfrak p$, relevant for superspace formulations of $11$-dimensional supergravity in terms of nonholonomic superstructures. This includes novel fermionic Spencer groups, providing with new cohomology classes of $\mathbb Z$-grading $1$ and form number $2$. Using the Hilbert-Poincaré series and the Euler characteristic, we also explore Spencer cohomology contributions in higher form numbers. We then propose a new general definition of filtered deformations of graded Lie superalgebras along first-order fermionic directions and investigate such deformations of $\mathfrak p$ that are maximally supersymmetric. In particular, we establish a no-go type theorem for maximally supersymmetric filtered subdeformations of $\mathfrak p$ along timelike (i.e., generic) first-order fermionic directions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_16869 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fermionic Spencer Cohomologies of D=11 Supergravity Cremonini, C. A. Grassi, P. A. Noris, R. Ravera, L. Santi, A. High Energy Physics - Theory Mathematical Physics Representation Theory We combine the theory of Cartan-Tanaka prolongations with the Molien-Weyl integral formula and Hilbert-Poincaré series to compute the Spencer cohomology groups of the $D=11$ Poincaré superalgebra $\mathfrak p$, relevant for superspace formulations of $11$-dimensional supergravity in terms of nonholonomic superstructures. This includes novel fermionic Spencer groups, providing with new cohomology classes of $\mathbb Z$-grading $1$ and form number $2$. Using the Hilbert-Poincaré series and the Euler characteristic, we also explore Spencer cohomology contributions in higher form numbers. We then propose a new general definition of filtered deformations of graded Lie superalgebras along first-order fermionic directions and investigate such deformations of $\mathfrak p$ that are maximally supersymmetric. In particular, we establish a no-go type theorem for maximally supersymmetric filtered subdeformations of $\mathfrak p$ along timelike (i.e., generic) first-order fermionic directions. |
| title | Fermionic Spencer Cohomologies of D=11 Supergravity |
| topic | High Energy Physics - Theory Mathematical Physics Representation Theory |
| url | https://arxiv.org/abs/2411.16869 |