Understand
We test a recently proposed wall-crossing formula for the change of the Hilbert space of BPS states in d=4,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 Goettsche 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.