箱图与奇异纤维
Box Graphs and Singular Fibers
摘要：
通过研究三维N = 2的含物质超对称规范理论，确定了椭圆纤维型四重态的高余维纤维，该规范理论描述了在四重态奇异模型Weierstrass模型上紧化的M理论的低能有效理论.这个理论的库仑分支的每一个阶段对应于一个特定的分辨率的维尔斯特拉斯模型，我们表明，这些有一个简洁的描述装饰盒图的基础上的表示图的物质多重态，或者由一类凸路径上说，图。相位之间的转换有一个简单的解释为路径的“翻转”，在几何形状中对应于实际的翻转转换。这种相的描述使我们能够枚举和确定它们之间的整个网络，并为所有还原李群提供各种物质表示。此外，我们观察到，每个网络的相位进行一个特定的李代数的（准）极小表示的结构。从几何的角度解释，这种分析确定了有效曲线的锥的生成元，以及奇异椭圆Calabi-Yau四重曲线的crepant分辨率之间的翻转转换网络。从箱形图中，我们确定了余维2和余维3中的所有纤维类型，并且我们发现了E_6、E_7和E_8的新的非小平纤维类型。
英文摘要：
We determine the higher codimension fibers of elliptically fibered Calabi-Yau fourfolds with section by studying the three-dimensional N=2 supersymmetric gauge theory with matter which describes the low energy effective theory of M-theory compactified on the associated Weierstrass model, a singular model of the fourfold. Each phase of the Coulomb branch of this theory corresponds to a particular resolution of the Weierstrass model, and we show that these have a concise description in terms of decorated box graphs based on the representation graph of the matter multiplets, or alternatively by a class of convex paths on said graph. Transitions between phases have a simple interpretation as `flopping' of the path, and in the geometry correspond to actual flop transitions. This description of the phases enables us to enumerate and determine the entire network between them, with various matter representations for all reductive Lie groups. Furthermore, we observe that each network of phases carries the structure of a (quasi-)minuscule representation of a specific Lie algebra. Interpreted from a geometric point of view, this analysis determines the generators of the cone of effective curves as well as the network of flop transitions between crepant resolutions of singular elliptic Calabi-Yau fourfolds. From the box graphs we determine all fiber types in codimensions two and three, and we find new, non-Kodaira, fiber types for E_6, E_7 and E_8.
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