Counting BPS Solitons and Applications

We propose a novel and simple method of computing the volume of the moduli space of BPS solitons in supersymmetric gauge theory. We use a D-brane realization of vortices and T-duality relation to domain walls. We there use a special limit where domain walls reduce to gas of hard (soft) one-dimensional rods for the Abelian (non-Abelian) cases. In the simpler cases of the Abelian-Higgs model on a torus, our results agree with exact results which are geometrically derived by an explicit integration over the moduli space of the vortices. On the other side of the limit, we can compute the volume of the moduli space in the combinatorial way, where the problem on the random (plane) partition appears as well as the four dimensional instanton calculus. A part of this talk is based on collaboration with M. Eto, T. Fujimori, M. Nitta, K. Ohashi and N. Sakai [hep-th/0703197].