Wall-crossing, Toric divisor and Seiberg duality

Abstract

We study the wall-crossing phenomena of BPS D4-D2-D0 states on the conifold and orbifold $ {{{{{\mathbb{C}}^2}}} \left/ {{{{\mathbb{Z}}_2}}} \right.} $ , from the viewpoint of the quiver quantum mechanics on the D-branes. The Kahler moduli dependence of the BPS index is translated into the FI parameter dependence of the Witten index. The wall-crossing phenomena are related to the Seiberg dualities of the quiver quantum mechanics. All the differences from the D6-D2-D0 case arise from the additional superpotential and “anti-quark” induced by the D4-brane. When the D-branes are on the conifold, the flop transition changes the duality cascade. When the D-branes are on the orbifold $ {{{{{\mathbb{C}}^2}}} \left/ {{{{\mathbb{Z}}_2}}} \right.} $ , the generating function of the Witten index is always given by a character of the affine SU(2) algebra. Both are consistent with the wall-crossing formula for BPS indices.