11D supergravity at ${\cal O}(l^3)$

Abstract

We compute certain spinorial cohomology groups controlling possible supersymmetric deformations of eleven-dimensional supergravity up to order $l^3$ in the Planck length. At ${\cal O}(l)$ and ${\cal O}(l^2)$ the spinorial cohomology groups are trivial and therefore the theory cannot be deformed supersymmetrically. At ${\cal O}(l^3)$ the corresponding spinorial cohomology group is generated by a nontrivial element. On an eleven-dimensional manifold $M$ such that $p_1(M)\neq 0$, this element corresponds to a supersymmetric deformation of the theory, which can only be redefined away at the cost of shifting the quantization condition of the four-form field strength.