论文标题
本地$ p^2 $的BPS树枝状镜检查
BPS Dendroscopy on Local $P^2$
论文作者
论文摘要
IIA型字符串理论中BPS状态的频谱在Calabi-yau上压缩了三倍，著名地跳过了复杂的Kähler模量空间中的Codimension-One壁，导致了复杂的腔室结构。分解吸引子的猜想认为，可以从吸引子索引$ω_*（γ_i）$从其各自的吸引人的室内的bps指数$ω_*（γ_I）$从吸引力的$γ_i$中重建BPS索引$ω_z_z（γ）$（γ）$可以从吸引子索引$ω_*（γ_I）中重建。如果正确，这将提供BPS光谱的分类（或树枝状镜）为嵌套BPS结合状态的不同拓扑，每个均具有简单的腔室结构。在这里，我们为最简单的，尽管是非紧密的，卡拉比yau三倍，即投射平面上的规范捆绑包$ p^2 $。由于Kähler模量空间具有复杂的维度，并且吸引子的流量保留了中心电荷的论点，因此吸引子的流动树与射线的散射序列相吻合，在散射图的二维切片中，在稳定条件的二维切片上，在派生的紧凑型连贯分层$ k_ $ k_ p^p^2} $ $ k_的派生类别类别上。我们将先前的结果结合在大容量切片中的$ k_ {p^2} $的散射图上，并在Orbifold Point $ \ Mathbb {C}^3/\ Mathbb {Z} _3 $附近的新结果结合在一起，并证明了分裂吸引人的流动构想对$π$ - 稳定性$π$ - 稳定性。 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 观点。
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.
