Finite energy Dirac-Born-Infeld monopoles and string junctions

The N = 4 supersymmetric Yang-Mills (SYM) theory with gauge group SU(k) spontaneously broken to U(1)k−1 has a spectrum of 1/2 supersymmetric magnetic monopoles and dyons which, together with the ‘elementary’ particles of the perturbative spectrum, fill out orbits of an SL(2; Z) electromagnetic duality group. Each such particle has an interpretation in IIB superstring theory as an (m, n) string stretched between a pair of parallel D3-branes, chosen from among k parallel D3-branes. For k = 2 there are no other particles in the spectrum but for k ≥ 3 there are additional, 1/4 supersymmetric, dyons that are entirely non-perturbative in the sense that they belong to SL(2; Z) orbits that contain no ‘elementary’ particles. Although these can be found as classical solutions of the SYM field equations [1–3] they were first found as IIB superstring configurations in which three strings of different (m, n) charges, attached to three D3branes, meet at a string junction [4]. These are points at which two IIB strings of charges (m, n) and (m′, n′) meet to form a string of charge (m + m′, n + n′) [5]. The minimum energy state to which the configuration relaxes is one in which three strings leaving the three D3-branes meet at a planar string junction [6,7]. Actually, the effective action of the D3-branes is not a SYM theory but rather a supersymmetric non-abelian Dirac-Born-Infeld (DBI) theory. The precise nature of this theory is not known (see [8] for a recent discussion) but it has an expansion in powers of α′F that simplifies in certain limits; α′ is the inverse IIB string tension and F is the (background covariant) Born-Infeld field strength. If L is the minimal separation between the D3-branes then (as we shall later see explicitly) α′F ∼ L2/α′, so the expansion parameter is actually L2/α′. For L << √ α′ we need keep only the quadratic terms in F and the action reduces to the N = 4 SYM theory (for a vacuum IIB background). For L >> √ α′ we cannot truncate the expansion but we may neglect the non-abelian interactions; the action then reduces to a sum of abelian DBI actions governing the dynamics of independent parallel D3-branes. The D3-brane action depends on the supergravity background. For example, F = F −B where F is the usual 2-form U(1) field strength, satisfying dF = 0, and B is the pullback of the background NS-NS 2-form potential. The D3-brane couples to the background R-R gauge fields through a Wess-Zumino Lagrangian LWZ . Let ξ be the worldvolume coordinates, (i = 0, 1, 2, 3). Omitting fermions and setting α′ = 1, the Lagrangian is then