Reflexive polygons have been extensively studied in a variety of contexts in mathematics and physics. We generalize this programme by looking at the 45 different lattice polygons with two interior points up to SL(2,ℤ) equivalence. Each corresponds to some affine toric 3-fold as a cone over a Sasaki-Einstein 5-fold. We study the quiver gauge theories of D3-branes probing these cones, which coincide with the mesonic moduli space. The minimum of the volume function of the Sasaki-Einstein base manifold plays an important role in computing the R-charges. We analyze these minimized volumes with respect to the topological quantities of the compact surfaces constructed from the polygons. Unlike reflexive polytopes, one can have two fans from the two interior points, and hence give rise to two smooth varieties after complete resolutions, leading to an interesting pair of closely related geometries and gauge theories.

中文翻译：

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箭袋规范理论：超越自反性

自反多边形已在数学和物理学的各种背景下进行了广泛的研究。我们通过查看 45 个不同的格子多边形来概括这个程序，这些多边形具有两个内点，直到 SL(2,ℤ) 等价。每个对应于 Sasaki-Einstein 5 倍上的一些仿射复曲面 3 倍作为锥体。我们研究了 D3-膜探测这些锥体的颤动规范理论，这些锥体与介子模量空间重合。Sasaki-Einstein 基流形的体积函数的最小值在计算 R 电荷中起着重要作用。我们根据由多边形构造的紧凑表面的拓扑量来分析这些最小化体积。与自反多胞体不同，一个可以有两个来自两个内点的扇形，因此在完全分解后会产生两个平滑的变体，