Mehen, T; Wise, MB, *Generalized *-products, Wilson lines and the solution of the Seiberg-Witten equations*,
Journal of High Energy Physics, vol. 4 no. 12
(December, 2000),
pp. 8 .
**Abstract:**

*Higher order terms in the effective action of non-commutative gauge theories exhibit generalizations of the *-product (e.g. *′and *3). These terms do not manifestly respect the non-commutative gauge invariance of the tree level action. In U(1) gauge theories, we note that these generalized *-products occur in the expansion of some quantities that are invariant 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 tensor Fabin terms of the non-commutative gauge field Âa. Seiberg 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 Â but keeping all orders in the non-commutative parameter θkl.*