Ab Initio Wall-Crossing

TL;DR

The paper derives an ab initio, ${\cal N}=4$ quantum-mechanical description of $n$ BPS centers near a wall of marginal stability and shows that a naive reduction to the classical moduli space ${\cal M}_n$ fails due to absence of scale separation. By partially breaking supersymmetry to ${\cal N}=1$ in a way that preserves the index, the authors obtain a Dirac index on ${\cal M}_n$ that counts non-threshold bound states, and they relate this to the field-theory protected spin character via a diagonal ${\cal J}_3=J_3+I_3$ symmetry. Incorporating Bose/Fermi statistics and using the MPS framework, they derive the general wall-crossing formula and reveal the universal rational invariant ${\bar\Omega}(\gamma)=\sum_{p|\gamma}{\Omega^+(\gamma/p)}/{p^2}$ as a natural outcome of fixed-submanifold contributions. The work also connects to orbifold Dirac indices and equivariant indices, providing alternative derivations and clarifying the relationship between field theory and quantum-mechanical state counts. Overall, the results unify ab initio derivations of Coulomb-phase wall-crossing with the MPS formulation and clarify the role of rational invariants in BPS spectrum transitions.

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 ~Ω(β)/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 β. Along the way, we also clarify relationships between field theory state countings and quantum mechanical indices.