论文标题
Brill-Noether General Limim Limit Bund Bundles：在F理论标准模型中不存在类似向量的Exotics
Brill-Noether-general Limit Root Bundles: Absence of vector-like Exotics in F-theory Standard Models
论文作者
论文摘要
在4D理论压缩的矢量样光谱研究中，根束显着。对现象学的重要性特别重要的是四亿f理论标准模型（F理论QSM）。在这项工作中，我们分析了物理根捆的超集，其共同体的共同体编码物质表示的矢量样光谱$（\ Mathbf {3}，\ Mathbf {2}）_ {1/6} $（1/6} $，$（ $（\ Mathbf {1}，\ Mathbf {1}）_ {1} $。对于家庭$ b_3（Δ_4^\ circ）$由$ \ mathcal {o}组成（10^{11}）$ f-theory QSM几何形状，我们认为超过$ 99.995 \％\％的根源中的根源没有vector-vector类似的exotics。这表明在这些表示中缺乏类似矢量的外来者是很可能的情况。 QSM几何形式来自从某些三维多型$δ^\ circ $的三角剖分中获得的三个折叠式$ b_3（δ^\ circ）$的家族。物质曲线在b_3中的$x_σ\（δ^\ circ）$可以变形为节点曲线，对于$ b_3（δ^\ circ）$中的所有空间都相同。因此，可以从几个淋巴结曲线的研究中探测整个家族$ b_3（δ^\ circ）$的矢量样光谱。我们计算这些淋巴结曲线上所有极限根的共同体。 在我们的应用中，对于大多数极限根，共同体是由线束的共同体在理性树状曲线上确定的。为此，我们提出了一种计算机算法。剩余的极限根（对应于电路样图形）是用手处理的。共同体独立于节点的相对位置，除了几个电路。在这些\ emph {跳跃电路}上，如果节点特别对齐，则可以跳线共同体。这反映了古典的Brill-Noether跳跃。 $ b_3（δ_4^\ circ）$允许跳跃电路，但是根束约束选择规范束，没有跳跃。
Root bundles appear prominently in studies of vector-like spectra of 4d F-theory compactifications. Of particular importance to phenomenology are the Quadrillion F-theory Standard Models (F-theory QSMs). In this work, we analyze a superset of the physical root bundles whose cohomologies encode the vector-like spectra for the matter representations $(\mathbf{3}, \mathbf{2})_{1/6}$, $(\mathbf{\overline{3}}, \mathbf{1})_{-2/3}$ and $(\mathbf{1}, \mathbf{1})_{1}$. For the family $B_3( Δ_4^\circ )$ consisting of $\mathcal{O}(10^{11})$ F-theory QSM geometries, we argue that more than $99.995\%$ of the roots in this superset have no vector-like exotics. This indicates that absence of vector-like exotics in those representations is a very likely scenario. The QSM geometries come in families of toric 3-folds $B_3( Δ^\circ )$ obtained from triangulations of certain 3-dimensional polytopes $Δ^\circ$. The matter curves in $X_Σ\in B_3( Δ^\circ )$ can be deformed to nodal curves which are the same for all spaces in $B_3( Δ^\circ )$. Therefore, one can probe the vector-like spectra on the entire family $B_3( Δ^\circ )$ from studies of a few nodal curves. We compute the cohomologies of all limit roots on these nodal curves. In our applications, for the majority of limit roots the cohomologies are determined by line bundle cohomology on rational tree-like curves. For this, we present a computer algorithm. The remaining limit roots, corresponding to circuit-like graphs, are handled by hand. The cohomologies are independent of the relative position of the nodes, except for a few circuits. On these \emph{jumping circuits}, line bundle cohomologies can jump if nodes are specially aligned. This mirrors classical Brill-Noether jumps. $B_3( Δ_4^\circ )$ admits a jumping circuit, but the root bundle constraints pick the canonical bundle and no jump happens.
