Crossing the wall: branes versus bundles

Abstract

We test a recently proposed wall-crossing formula for the change of the Hilbert space of Bogomol’nyi–Prasad–Sommerfield (BPS) states in $d = 4, \mathcal{N} = 2$ theories. We study decays of $D4D2D0$ systems into pairs of $D4D2D0$ systems and we show how the wall-crossing formula reproduces results of Göttsche and Yoshioka on wall-crossing behavior of the moduli of slope-stable holomorphic bundles over holomorphic surfaces. Our comparison shows very clearly that the moduli space of the $D4D2D0$ system on a rigid surface in a Calabi–Yau is not the same as the moduli space of torsion-free sheaves, even when worldhseet instantons are neglected. Moreover, we argue that the physical formula should make some new mathematical predictions for a future theory of the moduli of stable objects in the derived category.