*-Trek II: *n operations, open Wilson lines and the Seiberg–Witten map Original Research Article

Generalizations of the ∗-product (e.g., n-ary ∗n operations) appear in various places in the discussion of noncommutative gauge theories. These include the one-loop effective action of noncommutative gauge theories, the couplings between massless closed and open string modes, and the Seiberg–Witten map between the ordinary and noncommutative Yang–Mills fields. We propose that the natural way to understand the ∗n operations is through the expansion of an open Wilson line. We establish the connection between an open Wilson line and the ∗n operations and use it to (I) write down a gauge invariant effective action for the one-loop F4 terms in the noncommutative N=4 SYM theory; (II) find the gauge invariant couplings between the noncommutative SYM modes and the massless closed string modes in flat space; (III) propose a closed form for the Seiberg–Witten map in the U(1) case.