D=10 super-Yang-Mills at O(alpha '(2))

Using superspace techniques, the complete and most general action of D = 10 super-Yang-Mills theory is constructed at the alpha'(2) level. No other ap proximations, e.g., keeping only a subset of the allowed derivative terms, are used. The Lorentz structure of the alpha'(2) corrections is completely determined, while (depending on the gauge group) there is some freedom in t he adjoint structure, which is given by a totally symmetric four-index tens or. We examine the second, non-linearly realised supersymmetry that may be present when the gauge group has a U(1) factor, and find that the constrain ts from linear and non-linear supersymmetry to a large extent coincide. How ever, the additional restrictions on the adjoint structure of the order alp ha'(2) interactions following from the requirement of non-linear supersymme try do not completely specify the symmetrised trace prescription.