Crossing the Wall: Branes vs. Bundles

TL;DR

This paper tests a wall-crossing formula for BPS states in 4D N=2 theories, showing its consistency with known mathematical results and predicting new insights into moduli spaces of stable objects.

Contribution

It demonstrates the applicability of a recent wall-crossing formula to D-brane systems and reveals differences between physical and mathematical moduli spaces, suggesting new mathematical predictions.

Findings

Wall-crossing formula reproduces known results on moduli of stable bundles.

Moduli space of D4D2D0 systems differs from that of torsion-free sheaves.

Physical predictions may inform future mathematical theories of derived category 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.