Localization method for volume of domain-wall moduli spaces

The volume of the moduli space of non-Abelian Bogomol'nyi-Prasad-Sommerfield (BPS) domain walls is obtained exactly in gauge theory with N f matters. The volume of the moduli space is formulated, without an explicit metric, by a path integral under constraints on BPS equations. The path integral over fields reduces to a finite-dimensional contour integral by a localization mechanism. Our volume formula satisfies a Seiberg-like duality between moduli spaces of the and non-Abelian BPS domain walls in a strong coupling region. We also find a T-duality between domain walls and vortices on a cylinder. The moduli space volume of non-Abelian local ( ) vortices on the cylinder agrees exactly with that on a sphere. The volume formula reveals various geometrical properties of the moduli space.