*-Trek: the one-loop N=4 noncommutative SYM action

We investigate N=4 noncommutative super-Yang-Mills (SYM) theory. We compute the one-loop four gauge boson scattering amplitude on parallel Dp-branes, and find the corresponding contribution to the noncommutative SYM one-loop action in a momentum expansion. The result is somewhat surprising. We find that while the planar diagram can be written using the usual *-product, the contributions from nonplanar diagrams in general involve additional structure beyond the *-product, arising from the nontrivial worldsheet correlations surviving the field theory limit. To each nonplanar diagram, depending on the number n of external vertex operator insertions on each boundary, there is a corresponding *n n-ary operation. We further find that it is no longer possible to write down an off-shell gauge invariant one-loop effective action using the noncommutative field strength defined at tree-level