Fermionic Spencer Cohomologies of D=11 Supergravity

Abstract

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.

v2: 40 pages, final version to appear on Adv. Theor. Math. Phys