An analog of bulk-brane duality in field theory

Abstract

We consider ($3+1$)-dimensional $\mathcal{N}=2$ supersymmetric QED with two flavors of fundamental hypermultiplets. This theory supports $1/2$-Bogomol'nyi-Prasad-Sommerfield (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 ($W\overline{W}$) on a cylinder. We argue that for certain values of parameters this setup provides a controllable analog of bulk-brane duality in field theory. Dynamics of the 4D bulk are mapped onto 3D boundary theory: 3D $\mathcal{N}=2$ supersymmetric quantum electrodynamics (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 $\mathcal{N}=2$ 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.