Inspired by the split attractor flow conjecture for multi-centered black hole solutions in N=2 supergravity, we propose a formula expressing the BPS index $\Omega(\gamma,z)$ in terms of `attractor indices' $\Omega_*(\gamma_i)$. The latter count BPS states in their respective attractor chamber. This formula expresses the index as a sum over stable flow trees weighted by products of attractor indices. We show how to compute the contribution of each tree directly in terms of asymptotic data, without having to integrate the attractor flow explicitly. Furthermore, we derive new representations for the index which make it manifest that discontinuities associated to distinct trees cancel in the sum, leaving only the discontinuities consistent with wall-crossing. We apply these results in the context of quiver quantum mechanics, providing a new way of computing the Betti numbers of quiver moduli spaces, and compare them with the Coulomb branch formula, clarifying the relation between attractor and single-centered indices.

中文翻译：

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吸引子流树、BPS 指数和箭袋

受 N=2 超引力下多中心黑洞解的分裂吸引子流猜想的启发，我们提出了一个公式来表达 BPS 指数 $\Omega(\gamma,z)$ 的“吸引子指数”$\Omega_*( \gamma_i)$。后者在其各自的吸引器室中计算 BPS 状态。该公式将指数表示为由吸引子指数的乘积加权的稳定流树的总和。我们展示了如何直接根据渐近数据计算每棵树的贡献，而无需显式地整合吸引子流。此外，我们为索引导出了新的表示，这表明与不同树相关的不连续性在总和中抵消，只留下与穿墙一致的不连续性。我们将这些结果应用于颤抖量子力学的背景下，