Wall-Crossing from Boltzmann Black Hole Halos

TL;DR

The paper unifies the physics of BPS wall-crossing in N=2 theories with deep mathematical structures by representing bound states as multi-centered black holes and recasting the problem in two complementary frameworks: a Higgs-branch quiver approach and a Coulomb-branch localization approach. It introduces a Boltzmannian reinterpretation with effective rational invariants $\bar{\Omega}$ (and refined $\bar{\Omega}_{ref}$), which makes charge conservation manifest and enables general semi-primitive and higher-body wall-crossing generalizations. The authors derive explicit Higgs-branch and Coulomb-branch formulae, prove their agreement with Kontsevich-Soibelman and Joyce-Song wall-crossing formulæ in many cases, and extend the analysis to motivic/integer refinements, obtaining comprehensive consistency checks. Together, these results provide a physically intuitive and computationally efficient framework that connects supergravity, quiver quantum mechanics, and the mathematical theory of Donaldson-Thomas invariants, with broad implications for BPS spectra in both string theory and gauge theory contexts.

Abstract

A key question in the study of N=2 supersymmetric string or field theories is to understand the decay of BPS bound states across walls of marginal stability in the space of parameters or vacua. By representing the potentially unstable bound states as multi-centered black hole solutions in N=2 supergravity, we provide two fully general and explicit formulae for the change in the (refined) index across the wall. The first, "Higgs branch" formula relies on Reineke's results for invariants of quivers without oriented loops, specialized to the Abelian case. The second, "Coulomb branch" formula results from evaluating the symplectic volume of the classical phase space of multi-centered solutions by localization. We provide extensive evidence that these new formulae agree with each other and with the mathematical results of Kontsevich and Soibelman (KS) and Joyce and Song (JS). The main physical insight behind our results is that the Bose-Fermi statistics of individual black holes participating in the bound state can be traded for Maxwell-Boltzmann statistics, provided the (integer) index Ω(γ) of the internal degrees of freedom carried by each black hole is replaced by an effective (rational) index \barΩ(γ)= \sum_{m|γ} Ω(γ/m)/m^2. A similar map also exists for the refined index. This observation provides a physical rationale for the appearance of the rational Donaldson-Thomas invariant \barΩ(γ) in the works of KS and JS. The simplicity of the wall crossing formula for rational invariants allows us to generalize the "semi-primitive wall-crossing formula" to arbitrary decays of the type γ\to Mγ_1+Nγ_2 with M=2,3.

Figures (1)

• Figure 1: The 16 labelled trees contributing to $\mathcal{L}(\alpha_1,\alpha_2,\alpha_3,\alpha_4)$.