Generalized *-products, Wilson lines and the solution of the Seiberg-Witten equations

Higher order terms in the effective action of non-commutative gauge theorie s exhibit generalizations of the star -product (e.g. star' and star (3)). T hese terms do not manifestly respect the non-commutative gauge invariance o f the tree level action. In U(1) gauge theories, we note that these general ized star -products occur in the expansion of some quantities that are inva riant under non-commutative gauge transformations, but contain an infinite number of powers of the non-commutative gauge field. One example is an open Wilson line. Another is the expression for a commutative field strength te nsor F-ab in terms of the non-commutative gauge field (A) over cap (a). Sie berg and Witten derived differential equations that relate commutative and non-commutative gauge transformations, gauge fields and field strengths. In the U(1) case we solve these equations neglecting terms of fourth order in (A) over cap but keeping all orders in the non-commutative parameter theta (kl).