On higher-order corrections in M theory

A theoretical analysis of higher-order corrections to D=11 supergravity isgiven in a superspace framework. It is shown that any deformation of D=11supergravity for which the lowest-dimensional component of the four-form $G_4$vanishes is trivial. This implies that the equations of motion of D=11supergravity are specified by an element of a certain spinorial cohomologygroup and generalises previous results obtained using spinorial or pure spinorcohomology to the fully non-linear theory. The first deformation of the theoryis given by an element of a different spinorial cohomology group withcoefficients which are local tensorial functions of the massless supergravityfields. The four-form Bianchi Identities are solved, to first order and atdimension $-{1/2}$, in the case that the lowest-dimensional component of $G_4$is non-zero. Moreover, it is shown how one can calculate the first-ordercorrection to the dimension-zero torsion and thus to the supergravity equationsof motion given an explicit expression for this object in terms of thesupergravity fields. The version of the theory with both a four-form and aseven-form is discussed in the presence of the five-brane anomaly-cancellingterm. It is shown that the supersymmetric completion of this term exists and itis argued that it is the unique anomaly-cancelling invariant at this dimensionwhich is at least quartic in the fields. This implies that the firstdeformation of the theory is completely determined by the anomaly term fromwhich one can, in principle, read off the corrections to all of the superspacefield strength tensors.Comment: 32 pages. v2: Two references added in the text; footnote adde