Bulk–Brane Duality in Field Theory

We consider (3+1)-dimensional N = 2 supersymmetric QED with two flavors of fundamental hypermultiplets. This theory supports 1/2-BPS domain walls and flux tubes (strings), as well as their 1/4-BPS junctions. The effective (2+1)-dimensional theory on the domain wall is known to be a U(1) gauge theory. Previously, the wall–string junctions were shown to play the role of massive charges in this theory. However, the field theory of the junctions on the wall (for semi–infinite strings) appears to be inconsistent due to infrared problems. All these problems can be eliminated by compactifying one spatial dimension orthogonal to the wall and considering a wall–antiwall system (WW̄ ) on a cylinder. We argue that for certain values of parameters this set-up provides an example of a controllable bulk–brane duality in field theory. Dynamics of the 4D bulk are mapped onto 3D boundary theory: 3D N = 2 SQED with two matter superfields and a weak–strong coupling constant relation in 4D and 3D, respectively. The cylinder radius is seen as a “real mass” in 3D N = 2 SQED. We work out (at weak coupling) the quantum version of the world-volume theory on the walls. Integrating out massive matter (strings in the bulk theory) one generates a Chern–Simons term on the wall world volume and an interaction between the wall and antiwall that scales as a power of distance. Vector and scalar (classically) massless excitations on the walls develop a mass gap at the quantum level; the long-range interactions disappear. The above duality implies that the wall and its antiwall partner (at strong coupling in the bulk theory) are stabilized at the opposite sides of the cylinder.