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

Higher order terms in the effective action of noncommutative gauge theoriesexhibit generalizations of the *-product (e.g. *' and *-3). These terms do notmanifestly respect the noncommutative gauge invariance of the tree levelaction. In U(1) gauge theories, we note that these generalized *-products occurin the expansion of some quantities that are invariant under noncommutativegauge transformations, but contain an infinite number of powers of thenoncommutative gauge field. One example is an open Wilson line. Another is theexpression for a commutative field strength tensor in terms of thenoncommutative gauge field. Seiberg and Witten derived differential equationsthat relate commutative and noncommutative gauge transformations, gauge fieldsand field strengths. In the U(1) case we solve these equations neglecting termsof fourth order in the gauge field but keeping all orders in the noncommutativeparameter.Comment: 10 pages, minor changes to text, references adde