Supersymmetric gauge theory on the graph
Supersymmetric gauge theory on the graph
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图上的超对称规范理论
DOI：
10.1093/ptep/ptac018
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发表时间：
2022
影响因子：
3.5
通讯作者：
Ohta Kazutoshi
中科院分区：
文献类型：
--
作者：
Matsuura So;Ohta Kazutoshi
本文考虑二维N=（2，2）超对称规范理论在离散黎曼曲面上的应用。我们发现，离散化的理论可以有效地描述使用图论，其中玻色子和费米子场被视为一个图和它的对偶向量。我们首先分析阿贝尔理论，并确定其频谱的图论。特别地，我们证明了费米子具有对应于图的拓扑的零模，这可以理解为图和对偶图的关联矩阵的核。在连续理论中，标量曲率出现在与U（1）对称性相关的Ward-Takahashi（WT）恒等式中。我们发现，同样的异常出现的赤字角在图上的每个顶点。利用局部化方法，我们证明了图上的路径积分可化为零模集上的积分。因此，除非插入合适的算子，否则配分函数是病态的。我们把同样的结论推广到非阿贝尔理论，证明了在局部化不动点上路径积分可化为阿贝尔理论的重积分。
We consider two-dimensional N=(2,2) supersymmetric gauge theory on discretized Riemann surfaces. We find that the discretized theory can be efficiently described by using graph theory, where the bosonic and fermionic fields are regarded as vectors on a graph and its dual. We first analyze the Abelian theory and identify its spectrum in terms of graph theory. In particular, we show that the fermions have zero modes corresponding to the topology of the graph, which can be understood as kernels of the incidence matrices of the graph and the dual graph. In the continuous theory, a scalar curvature appears as an anomaly in the Ward-Takahashi (WT) identity associated with a U(1) symmetry. We find that the same anomaly arises as the deficit angle at each vertex on the graph. By using the localization method, we show that the path integral on the graph reduces to an integral over a set of the zero modes. The partition function is then ill-defined unless suitable operators are inserted. We extend the same argument to the non-Abelian theory and show that the path integral reduces to multiple integrals of Abelian theories at the localization fixed points.