BPS Dendroscopy on Local $P^2$
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arXiv
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2022
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| author | Bousseau, Pierrick Descombes, Pierre Floch, Bruno Le Pioline, Boris |
| author_facet | Bousseau, Pierrick Descombes, Pierre Floch, Bruno Le Pioline, Boris |
| contents | 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 $Ω_z(γ)$ for given charge $γ$ and moduli $z$ can be reconstructed from the attractor indices $Ω_*(γ_i)$ counting BPS states of charge $γ_i$ 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 $Π$-stability conditions. In particular, while there is an infinite set of initial rays related by the group $Γ_1(3)$ of auto-equivalences, only a finite number of possible decompositions $γ=\sum_iγ_i$ contribute to the index $Ω_z(γ)$ for any $γ$ and $z$, with constituents $γ_i$ related by spectral flow to the fractional branes at the orbifold point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_10712 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | BPS Dendroscopy on Local $P^2$ Bousseau, Pierrick Descombes, Pierre Floch, Bruno Le Pioline, Boris High Energy Physics - Theory Algebraic Geometry 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 $Ω_z(γ)$ for given charge $γ$ and moduli $z$ can be reconstructed from the attractor indices $Ω_*(γ_i)$ counting BPS states of charge $γ_i$ 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 $Π$-stability conditions. In particular, while there is an infinite set of initial rays related by the group $Γ_1(3)$ of auto-equivalences, only a finite number of possible decompositions $γ=\sum_iγ_i$ contribute to the index $Ω_z(γ)$ for any $γ$ and $z$, with constituents $γ_i$ related by spectral flow to the fractional branes at the orbifold point. |
| title | BPS Dendroscopy on Local $P^2$ |
| topic | High Energy Physics - Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2210.10712 |