Integral invariants in maximally supersymmetric Yang-Mills theories

Integral invariants in maximally supersymmetric Yang-Mills theories are discussed in spacetime dimensions 4 ≤ D ≤ 10 for SU(k) gauge groups. It is shown that, in addition to the action, there are three special invariants in all dimensions. Two of these,the single- and double-trace F⁴ invariants, are of Chern-Simons type in D = 9, 10 and BPS type in D ≤ 8, while the third, the double-trace of two derivatives acting on F⁴, can be expressed in terms of a gauge-invariant super-D-form in all dimensions. We show that the super-ten-forms for D = 10 F⁴ invariants have interesting cohomological properties and we also discuss some features of other invariants, including the single-trace d²F⁴, which has a special form in D = 10. The implications of these results for ultra-violet divergences are discussed in the framework of algebraic renormalisation.