BPS Dendroscopy on Local $P^2$

TL;DR

This paper verifies the Split Attractor Flow Conjecture for a specific non-compact Calabi-Yau threefold, demonstrating how BPS spectra can be classified via attractor flow trees and scattering diagrams in a simplified setting.

Contribution

It proves the conjecture for the canonical bundle over P^2, connecting scattering diagrams with BPS state classification in a concrete example.

Findings

Confirmed the conjecture on the physical slice of stability conditions.

Finite decompositions of charges contribute to BPS indices despite infinite initial rays.

Connected attractor flow trees with scattering sequences in the derived category.

Abstract

The spectrum of BPS states in type IIA string theory compactified on a Calabi-Yau threefold famously jumps across codimension-one walls in complexified K\"ahler 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 $\Omega_*(\gamma_i)$ 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 the projective plane $P^2$. Since the K\"ahler 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 in the space of stability conditions on the derived category of compactly supported coherent sheaves on $K_{P^2}$. We combine previous results on the scattering diagram of $K_{P^2}$ in the large volume slice with new results near 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.