Localization method for volume of domain-wall moduli spaces

摘要

The volume of the moduli space of non-Abelian Bogomol'nyi–Prasad–Sommerfield (BPS) domain walls is obtained exactly in $U(N_c)$ 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 $U(N_c)$ and $U(N_f-N_c)$ 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 ($N_c=N_f$) vortices on the cylinder agrees exactly with that on a sphere. The volume formula reveals various geometrical properties of the moduli space.