Mehen, T; Wise, MB, *Generalized *-Products, Wilson Lines and the Solution of the Seiberg-Witten Equations*,
JHEP, vol. 12 no. 12
(2000),
pp. 8 .
**Abstract:**

*Higher order terms in the effective action of noncommutative
gauge theories exhibit generalizations of the *-product
(e.g. *' and *-3). These terms do not manifestly respect the
noncommutative 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 noncommutative gauge transformations,
but contain an infinite number of powers of the
noncommutative gauge field. One example is an open Wilson
line. Another is the expression for a commutative field
strength tensor in terms of the noncommutative gauge field.
Seiberg and Witten derived differential equations that
relate commutative and noncommutative gauge transformations,
gauge fields and field strengths. In the U(1) case we solve
these equations neglecting terms of fourth order in the
gauge field but keeping all orders in the noncommutative
parameter.*