Ab Initio Wall-Crossing

Abstract

We derive supersymmetric quantum mechanics of n BPS objects with 3n position degrees of freedom and 4n fermionic partners with SO(4) R-symmetry. The potential terms, essential and sufficient for the index problem for non-threshold BPS states, are universal, and 2(n-1) dimensional classical moduli spaces M_n emerge from zero locus of the potential energy. We emphasize that there is no natural reduction of the quantum mechanics to M_n, contrary to the conventional wisdom. Nevertheless, via an index-preserving deformation that breaks supersymmetry partially, we derive a Dirac index on M_n as the fundamental state counting quantity. This rigorously fills a missing link in the "Coulomb phase" wall-crossing formula in literature. We then impose Bose/Fermi statistics of identical centers, and derive the general wall-crossing formula, applicable to both BPS black holes and BPS dyons. Also explained dynamically is how the rational invariant ~\Omega(\beta)/p^2, appearing repeatedly in wall-crossing formulae, can be understood as the universal multiplicative factor due to p identical, coincident, yet unbound, BPS particles of charge \beta. Along the way, we also clarify relationships between field theory state countings and quantum mechanical indices.