BPS Dendroscopy on Local $P^2$

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 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 ${\Omega }_{\ast }({\gamma }_i)$ 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 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 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.

arXiv comment61+23 pages, 32 figures; v3: final version to appear in Comm. Math. Phys

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Cited by in corpus (3)

• Reading between the rational sections: Global structures of 4d $\mathcal{N}=2$ KK theories
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