Fermionic Spencer Cohomologies of D=11 Supergravity

TL;DR

This paper computes the Spencer cohomology groups of the D=11 Poincaré superalgebra, revealing new fermionic cohomology classes and exploring deformations relevant for supergravity formulations.

Contribution

It introduces a novel method combining Cartan-Tanaka prolongations with Molien-Weyl integrals to compute superalgebra cohomology, including new fermionic Spencer groups and a framework for filtered deformations.

Findings

Identified new fermionic Spencer cohomology classes in D=11 supergravity.

Established a no-go theorem for certain maximally supersymmetric deformations.

Developed a new approach for analyzing filtered deformations of graded Lie superalgebras.

Abstract

We combine the theory of Cartan-Tanaka prolongations with the Molien-Weyl integral formula and Hilbert-Poincar\'e series to compute the Spencer cohomology groups of the $D=11$ Poincar\'e 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-Poincar\'e 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.