图哈密顿量的重规范广群、对称性和退化性及其在陀螺线网络中的应用
Re-gauging groupoid, symmetries and degeneracies for Graph Hamiltonians and applications to the Gyroid wire network
摘要：
我们研究了一类图的哈密尔顿算子给出的一种类型的表示，我们可以关联（非）-交换几何。通过选择测量数据，这些几何形状通过显式构造或Kan扩展由矩阵实现。我们描述的变化规范通过一个regauging广群的行动。它的行动通过矩阵，使产生一个非交换2-上循环，因此，以广群扩展（gerbe）。此外，我们表明，自同构的底层图的可提升的扩展对称群的regaugings。在交换的情况下，我们推导出的扩展对称行为通过一个投影表示。这就产生了同构分解和超选择规则。我们将这些结果应用到PDG和蜂窝线网络使用投影群的表示理论，并表明，所有的退化的光谱是这些增强的对称性的后果。这包括G（yroid）和蜂巢系统的狄拉克点。
英文摘要：
We study a class of graph Hamiltonians given by a type of quiver representation to which we can associate (non)--commutative geometries. By selecting gauging data these geometries are realized by matrices through an explicit construction or a Kan-extension. We describe the changes in gauge via the action of a regauging groupoid. It acts via matrices that give rise to a noncommutative 2--cocycle and hence to a groupoid extension (gerbe). We furthermore show that automorphisms of the underlying graph of the quiver can be lifted to extended symmetry groups of regaugings. In the commutative case, we deduce that the extended symmetries act via a projective representation. This yields isotypical decompositions and super--selection rules. We apply these results to the PDG and honeycomb wire--networks using representation theory for projective groups and show that all the degeneracies in the spectra are consequences of these enhanced symmetries. This includes the Dirac points of the G(yroid) and the honeycomb systems.
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