BPS Dendroscopy on Local \(\mathbb {P}^2\)

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.