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

TL;DR

This paper investigates possible supersymmetric deformations of eleven-dimensional supergravity at order $l^3$, revealing a nontrivial deformation linked to the manifold's topology that affects the four-form field strength quantization.

Contribution

It identifies a nontrivial supersymmetric deformation at order $l^3$ in supergravity, which is absent at lower orders, and connects this to topological properties of the underlying manifold.

Findings

No deformations at ${ m O}(l)$ and ${ m O}(l^2)$

A nontrivial deformation appears at ${ m O}(l^3)$

Deformation affects the four-form field strength quantization

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.