BPS Dendroscopy on Local $\mathbb{P}^1\times \mathbb{P}^1$

BPS states in type II string compactified on a Calabi-Yau threefold can typically be decomposed as moduli-dependent bound states of absolutely stable constituents, with a hierarchical structure labelled by attractor flow trees. This decomposition is best understood from the scattering diagram, an arrangement of real codimension-one loci (or rays) in the space of stability conditions where BPS states of given electromagnetic charge and fixed phase of the central charge exist. The consistency of the diagram when rays intersect determines all BPS indices in terms of the `attractor indices' carried by the initial rays. In this work we study the scattering diagram for a non-compact toric CY threefold known as local $\mathbb{F}_0$, namely the total space of the canonical bundle over $\mathbb{P}^1\times \mathbb{P}^1$. We first construct the scattering diagram for the quiver, valid near the orbifold point, and for the large volume slice, valid when both $\mathbb{P}^1$'s have large (and nearly equal) area. We then combine the insights gained from these simple limits to construct the scattering diagram along the physical slice of $\Pi$-stability conditions, which carries an action of a $\mathbb{Z}^4$ extension of the modular group $\Gamma_0(4)$. We sketch a proof of the Split Attractor Flow Tree Conjecture in this example, albeit for a restricted range of the central charge phase. Most arguments are similar to our early study of local $\mathbb{P}^2$ [arXiv:2210.10712], but complicated by the occurence of an extra mass parameter and ramification points on the $\Pi$-stability slice.