Supersymmetric deformations of maximally supersymmetric gauge theories

We study supersymmetric and super Poincaré invariant deformations of tendimensional super Yang-Mills theory and of its dimensional reductions. We describe all infinitesimal super Poincaré invariant deformations of equations of motion of ten-dimensional super Yang-Mills theory and deformations of the reduction to a point. We also discuss how these infinitesimals can be extended to formal deformations. Our methods are based on homological algebra, in particular, on the theory of L-infinity and A-infinity algebras. The exposition of this theory as well as of some basic facts about Lie algebra homology and Hochschild homology is given in appendices.