  The question of whether BPS invariants are protected in maximally
supersymmetric Yang-Mills theories is investigated from the point of view of
algebraic renormalisation theory. The protected invariants are those whose
cohomology type differs from that of the action. It is confirmed that one-half
BPS invariants ($F^4$) are indeed protected while the double-trace one-quarter
BPS invariant ($d^2F^4$) is not protected at two loops in D=7, but is protected
at three loops in D=6 in agreement with recent calculations. Non-BPS
invariants, i.e. full superspace integrals, are also shown to be unprotected.

High Energy Physics - Theory