Crossing the Wall: Branes vs. Bundles

TL;DR

The paper tests a universal wall-crossing formula for BPS state spaces in $d=4$, ${ m N}=2$ theories by examining $D4$-$D2$-$D0$ decays and comparing with Göttsche–Yoshioka results on moduli of slope-stable bundles. It demonstrates that, in the large Kähler limit, the physical wall-crossing matches mathematical predictions for moduli of coherent sheaves, but subleading corrections reveal that the true physical moduli space corresponds to stable objects in the derived category rather than mere sheaf moduli. This suggests a derived-category stability framework as the correct mathematical structure and yields new predictions for the behavior of stable derived objects under wall-crossing. The work also explores generalizations (across different surfaces, D6 decays, spectral-cover data) and discusses potential implications for OSV-type conjectures and future mathematical formulations of derived-stability moduli.

Abstract

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.