多面体上$\mathcal{N}=(2,2)$ SYM中的Anomaly和符号问题：数值分析
Anomaly and Sign problem in $\mathcal{N}=(2,2)$ SYM on Polyhedra : Numerical Analysis
摘要：
本文研究了离散曲面空间（多面体）上的二维$\mathcal{N}=(2,2)$超对称Yang-Mills（SYM）理论。我们首先回顾了连续$\mathcal{N}=(2,2)$ SYM理论在任何弯曲流形上的超对称性的数目可以通过引入与$U(1)_{V}$对称性相关联的适当的$U(1)$规范背景而至少增加到两个。然后，我们表明，广义Sugino模型的离散化弯曲空间，这是我们以前的工作中提出的，可以确定为这个超对称增强理论的离散化，其中一个超对称仍然存在，另一个是打破，但恢复在连续极限。我们发现，$U(1)_{A}$异常也存在于离散化理论中，作为一个不平衡的费米子的数量成比例的多面体的欧拉特征的结果。然后我们用数值蒙特卡罗模拟研究了这个模型。针对$U(1)_A$异常，我们提出了一种新的相位猝灭方法，称为“异常相位猝灭近似”。我们数值计算表明，Ward-Takahashi（WT）恒等式与剩余的超对称性是通过采用这种近似实现的。指出了符号（相位）问题与狄拉克算子的伪零模之间的关系。我们还发现，随着背景亏格的增加，标量一点函数的发散行为变得更温和。这是第一个数值观测的超对称晶格模型的弯曲空间与一般拓扑。
英文摘要：
We investigate the two-dimensional $\mathcal{N}=(2,2)$ supersymmetric Yang-Mills (SYM) theory on the discretized curved space (polyhedra). We first revisit that the number of supersymmetries of the continuum $\mathcal{N}=(2,2)$ SYM theory on any curved manifold can be enhanced at least to two by introducing an appropriate $U(1)$ gauge background associated with the $U(1)_{V}$ symmetry. We then show that the generalized Sugino model on the discretized curved space, which was proposed in our previous work, can be identified to the discretization of this SUSY enhanced theory, where one of the supersymmetries remains and the other is broken but restored in the continuum limit. We find that the $U(1)_{A}$ anomaly exists also in the discretized theory as a result of an unbalance of the number of the fermions proportional to the Euler characteristics of the polyhedra. We then study this model by using the numerical Monte-Carlo simulation. We propose a novel phase-quench method called "anomaly-phase-quenched approximation" with respect to the $U(1)_A$ anomaly. We numerically show that the Ward-Takahashi (WT) identity associated with the remaining supersymmetry is realized by adopting this approximation. We figure out the relation between the sign (phase) problem and pseudo-zero-modes of the Dirac operator. We also show that the divergent behavior of the scalar one-point function gets milder as the genus of the background increases. These are the first numerical observations for the supersymmetric lattice model on the curved space with generic topologies.
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