The non-abelian Born-Infeld action and noncommutative gauge theory

In this paper we explicitly show the equivalence between the non-abelian Born-Infeld action, which was proposed by Tseytlin as an effective action on several D-branes, and its noncommutative counterpart for slowly varying fields. This confirms the equivalence between the two descriptions of the D-branes using an ordinary gauge theory with a constant B field background and a noncommutative gauge theory, claimed by Seiberg and Witten. We also construct the general forms of the 2n-derivative terms for non-abelian gauge fields which are consistent with the equivalence in the approximation of neglecting (2n + 2)-derivative terms.