Spinorial cohomology and maximally supersymmetric theories

Fields in supersymmetric gauge theories may be seen as elements in a spinorial cohomology. We elaborate on this subject, specialising to maximally supersymmetric theories, where the superspace Bianchi identities, after suitable conventional constraints are imposed, put the theories on shell. In these cases, the spinorial cohomologies describe in a unified manner gauge transformations, fields and possible deformations of the models, e.g. string-related corrections in an alpha' expansion. Explicit cohomologies are calculated for super-Yang-Mills theory in D=10, for the N=(2,0) tensor multiplet in D=6 and for supergravity in D=11, in the latter case from the point of view of both the super-vielbein and the super-3-form potential. The techniques may shed light on some questions concerning the alpha'-corrected effective theories, and result in better understanding of the role of the 3-form in D=11 supergravity.