The spectrum of BPS states in type IIA string theory compactified on a Calabi–Yau threefold famously jumps across codimension-one walls in complexified Kähler moduli space, leading to an intricate chamber structure. The Split Attractor Flow Conjecture posits that the BPS index \(\Omega _z(\gamma )\) for given charge \(\gamma \) and moduli z can be reconstructed from the attractor indices \(\Omega _\star (\gamma _i)\) counting BPS states of charge \(\gamma _i\) in their respective attractor chamber, by summing over a finite set of decorated rooted flow trees known as attractor flow trees. If correct, this provides a classification (or dendroscopy) of the BPS spectrum into different topologies of nested BPS bound states, each having a simple chamber structure. Here we investigate this conjecture for the simplest, albeit non-compact, Calabi–Yau threefold, namely the canonical bundle over \(\mathbb {P}^2\). Since the Kähler moduli space has complex dimension one and the attractor flow preserves the argument of the central charge, attractor flow trees coincide with scattering sequences of rays in a two-dimensional slice of the scattering diagram \({\mathcal {D}}_\psi \) in the space of stability conditions on the derived category of compactly supported coherent sheaves on \(K_{\mathbb {P}^2}\). We combine previous results on the scattering diagram of \(K_{\mathbb {P}^2}\) in the large volume slice with an analysis of the scattering diagram for the three-node quiver valid in the vicinity of the orbifold point \(\mathbb {C}^3/\mathbb {Z}_3\), and prove that the Split Attractor Flow Conjecture holds true on the physical slice of \(\Pi \)-stability conditions. In particular, while there is an infinite set of initial rays related by the group \(\Gamma _1(3)\) of auto-equivalences, only a finite number of possible decompositions \(\gamma =\sum _i \gamma _i\) contribute to the index \(\Omega _z(\gamma )\) for any \(\gamma \) and z, with constituents \(\gamma _i\) related by spectral flow to the fractional branes at the orbifold point. We further explain the absence of jumps in the index between the orbifold and large volume points for normalized torsion free sheaves, and uncover new ‘fake walls’ across which the dendroscopic structure changes but the total index remains constant.

在卡拉比-丘三重上压缩的 IIA 型弦理论中的 BPS 态谱著名地跨越了复杂的凯勒模空间中的余维一壁，导致了复杂的腔室结构。分裂吸引子流猜想假设给定电荷\(\gamma \)和模z的 BPS 索引\(\Omega _z(\gamma )\)可以从吸引子索引\(\Omega _\star (\gamma _i)\)通过对一组有限的装饰根流树（称为吸引子流树）进行求和，计算各自吸引子室中的 BPS 电荷状态\(\gamma _i\) 。如果正确，这可以将 BPS 谱分类（或树状观察）为嵌套 BPS 束缚态的不同拓扑，每个拓扑都具有简单的室结构。在这里，我们研究最简单但非紧的 Calabi-Yau 三重猜想，即\(\mathbb {P}^2\)上的规范丛。由于凯勒模空间具有复数维度一，并且吸引子流保留了中心电荷的变角，因此吸引子流树与散射图的二维切片中的光线散射序列一致\({​​\mathcal {D}}_ \psi \)在\(K_{\mathbb {P}^2}\)上紧支撑相干滑轮的派生类别上的稳定条件空间中。我们将之前大体积切片中\(K_{\mathbb {P}^2}\)散射图的结果与对在轨道折叠点附近有效的三节点颤动的散射图分析相结合(\mathbb {C}^3/\mathbb {Z}_3\)，并证明分裂吸引子流猜想在\(\Pi \)稳定性条件的物理切片上成立。特别是，虽然存在由自等价组\(\Gamma _1(3)\)相关的无限组初始射线，但只有有限数量的可能分解\(\gamma =\sum _i \gamma _i\ )对任何\(\gamma \)和z 的索引\(\Omega _z(\gamma )\)做出贡献，其成分\(\gamma _i\)通过谱流与轨道折叠点处的分数膜相关。我们进一步解释了标准化无扭转滑轮的轨道折叠点和大体积点之间的指数没有跳跃，并发现了新的“假墙”，在这些“假墙”上，树状结构发生了变化，但总指数保持不变。