Instanton in Born-Infeld Action and Noncommutative Gauge Theory

We derive a BPS-type bound for four-dimensional Born-Infeld action with constant B field background. The supersymmetric configuration saturates this bound and is regarded as an analog of instanton in U(1) gauge theory. Furthermore, we find the explicit solutions of this BPS condition. These solutions have a finite action proportional to the instanton number and represent D(p−4)-branes within a Dp-brane although they have a singularity at the origin. Some relations to the noncommutative U(1) instanton are discussed. In recent development of string theory, it has been realized that noncommutative spacetime is naturally appeared in the D-brane physics. Indeed, it has been known that in a certain limit the transverse coordinates of D-branes can be regarded as matrices. This fact implies the non-commutativity of spacetime [1]. Furthermore the noncommutative Yang-Mills theory is realized on the world volume of D-branes with constant B field background. This has been shown in the context of Matrix theory in [2] and more directly considering open strings ending on the D-branes in [3, 4, 5]. In [11], Seiberg and Witten has promoted this idea deeper and argued that the noncommutative Yang-Mills theory is equivalent to the ordinary gauge theory. They have also discussed the relations between the ordinary instantons and the instantons in the noncommutative Yang-Mills theory [6, 7] which has no small instanton singularities. ∗ The instantons represent the D(p− 4)-branes within the Dp-brane. In the U(1) case, the effective action on the brane for the slowly varying fields has been known as the Dirac-Born-Infeld action [12]. It has been shown in [11] that the BPS condition of the ordinary Dirac-Born-Infeld action and a noncommutative one are equivalent in a limit α → 0 with an appropriate rescaling of the metric. The noncommutative U(1) gauge theory has instanton solutions [6] though the ordinary U(1) gauge theory can not has a nonsingular solutions with nonzero instanton numbers. In the limit, they have also constructed the BPS solutions with a finite instanton number and a singularity at the origin. In this paper we consider the BPS condition of the ordinary Dirac-Born-Infeld action with α fixed finite. We derive the BPS-type bound for the Born-Infeld action with constant B field background and show that the supersymmetric configuration indeed saturates this bound. Moreover we find the solutions of this BPS condition. These solutions have a finite action proportional to the integral of F ∧ F and may represent D(p − 4)-branes in a Dp-brane though they have a singularity at the origin. Although these solution can not be valid because of the singularity, we expect that the solutions is approximately valid away from the singularity. Thus the U(1) instanton solutions maybe teach us the relations between the two theories. ∗The noncommutative instanton on the torus has been discussed in [8]. The monopole in the noncommutative Yang-Mills theory has been considered in [9, 10]