On higher-order corrections in M theory

TL;DR

The paper investigates higher-order corrections to $D=11$ supergravity in a superspace framework and shows that deformations with $G_{0,4}=0$ are trivial, with the equations of motion encoded by the spinorial cohomology group $H^{0,3}_F$. The first nontrivial deformation arises from $G_{0,4} eq 0$, residing in $H^{0,4}_F(phys)$, and the four-form Bianchi identities fix the associated corrections to the torsion and the $G_4$ components; when a seven-form is included, anomaly cancellation selects a unique quartic-in-fields invariant that determines the first deformation. The authors provide a detailed perturbative solution of the $G_4$ Bianchi identities, derive the first-order torsion corrections, and demonstrate that the anomaly term governs the complete first deformation, enabling, in principle, the extraction of corrections to all superspace field-strength tensors. Collectively, the work offers a principled, cohomology-based route to computing $R^4$-type corrections in M-theory and clarifies how five-brane anomaly considerations fix the leading higher-order structure of the theory.

Abstract

A theoretical analysis of higher-order corrections to D=11 supergravity is given in a superspace framework. It is shown that any deformation of D=11 supergravity for which the lowest-dimensional component of the four-form $G_4$ vanishes is trivial. This implies that the equations of motion of D=11 supergravity are specified by an element of a certain spinorial cohomology group and generalises previous results obtained using spinorial or pure spinor cohomology to the fully non-linear theory. The first deformation of the theory is given by an element of a different spinorial cohomology group with coefficients which are local tensorial functions of the massless supergravity fields. The four-form Bianchi Identities are solved, to first order and at dimension $-{1/2}$, in the case that the lowest-dimensional component of $G_4$ is non-zero. Moreover, it is shown how one can calculate the first-order correction to the dimension-zero torsion and thus to the supergravity equations of motion given an explicit expression for this object in terms of the supergravity fields. The version of the theory with both a four-form and a seven-form is discussed in the presence of the five-brane anomaly-cancelling term. It is shown that the supersymmetric completion of this term exists and it is argued that it is the unique anomaly-cancelling invariant at this dimension which is at least quartic in the fields. This implies that the first deformation of the theory is completely determined by the anomaly term from which one can, in principle, read off the corrections to all of the superspace field strength tensors.