{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/bps-dendroscopy-on-local-mathbb-p-2","title":"BPS Dendroscopy on Local $P^2$","arxiv_id":"2210.10712","date":"2022-10-19","proceeding":null,"authors":["Pierrick Bousseau","Pierre Descombes","Bruno Le Floch","Boris Pioline"],"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.","url_abs":"https://arxiv.org/abs/2210.10712v3","url_pdf":"https://arxiv.org/pdf/2210.10712v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"bps-dendroscopy-on-local-mathbb-p-2","repo_url":"https://github.com/bpioline/p2scattering","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}