论文标题
几何形状的物质表示：在Dynkin的咒语下
Matter representations from geometry: under the spell of Dynkin
论文作者
论文摘要
在传统的katz-vafa方法中，物质表示形式是通过分解父母简单的lie代数$ \ mathfrak {m} $作为半密码subalgebra $ \ m athfrak {g} $的不可减至表示的直接总和。 $ \ mathfrak {m} $包含几个子代词同构为$ \ mathfrak {g} $，但是给出了邻接表示的不同分解，katz-vafa方法就会变得模棱两可。我们提出了一个选择规则，该规则表征了在F理论和M理论中观察到的物质表示形式：通用F理论紧凑型中的物质表示对应于subalgebras $ \ Mathfrak {g} \ subset \ subset \ subset \ mathfrak {m Mathfrak {m Mathfrak {m Mathfrak {m Mathfrak {m Mathfrak Index One in Dynkin Index One沿每个简单的组合$ $ nsible Compontents的线性等价类别。这个简单而优雅的选择规则使我们能够将Katz-Vafa方法应用于大型模型。我们说明了许多示例，该提案如何简化F理论中物质表示的推导并解决以前的歧义案例。
In the traditional Katz-Vafa method, matter representations are determined by decomposing the adjoint representation of a parent simple Lie algebra $\mathfrak{m}$ as the direct sum of irreducible representations of a semisimple subalgebra $\mathfrak{g}$. The Katz-Vafa method becomes ambiguous as soon as $\mathfrak{m}$ contains several subalgebras isomorphic to $\mathfrak{g}$ but giving different decompositions of the adjoint representation. We propose a selection rule that characterizes the matter representations observed in generic constructions in F-theory and M-theory: the matter representations in generic F-theory compactifications correspond to linear equivalence classes of subalgebras $\mathfrak{g}\subset \mathfrak{m}$ with Dynkin index one along each simple components of $\mathfrak{g}$. This simple yet elegant selection rule allows us to apply the Katz-Vafa method to a much large class of models. We illustrate on numerous examples how this proposal streamlines the derivation of matter representations in F-theory and resolves previously ambiguous cases.
