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