Construction of supersymmetric field theories by Ichimatsu-patterned lattice and application to nonperturbative renormalization group

本课题的目的是构建正则场论，以理解基本粒子物理的统一（超对称）模式，并获得正则场论实现对称性的新思路。由于我们的基本粒子物理学是由“规范原理”主导的，它的正则化版本也必须被测量。晶格规范理论是一种最合适的方法，但通常的立方晶格比我们的世界更不对称。一松模式晶格在正常的立方晶格内部有一个子结构。它是费米子自由度的特征。我们已经清楚地表明，一松图案晶格上的费米场是用SO（2D）而不是SO(D)来描述的，其中D是时空维度。这一令人惊讶的事实使我们理解了在单位晶格中费米子度是2^<(2D)/2>而不是2^<D/2>的加倍现象。超对称理论具有玻色子和费米子自由度之间的平衡，而规范自由度的对应物仍然是一个开放的问题。为什么实现超对称不像规范对称那么困难？其中一个原因是费米场的路径积分测量。最简单的情况是零维情况，我们研究了马约拉纳费米子存在下的全局异常。结果是，一致性理论强制我们在理论中有偶数个马约拉纳费米子，比如威滕的异常。另一方面，我们如何在晶格上设置马约拉纳费米子？根据Luescher关于手性对称的一种方法，我们尝试了一个Majorana条件。令人遗憾的是，手性汤川偶联的结果并不令人满意。因此，我们需要在正则场理论中得到系统的对称方法。量子主方程可以充分、必要地描述这一实现。这种方法可以应用于有外部引力场的系统。

The purpose of this project is to construct regularized field theories in order to understand a unified (supersymmetric) mode of fundamental particle physics and to obtain a new idea on symmetry realized in regularized field theories.Since our fundamental particle physics is dominated by “gauge principle", its regularized version must be also gauged. The lattice gauge theory is a most suitable approach but the usual cubic lattice is less symmetry than our world.An Ichimatsu-patterned lattice has a substructure inside the normal cubic lattice. It is characteristic on account of fermionic degrees of freedom. We have made it clear that fermi fields on a Ichimatsu-patterned lattice are described by SO(2D) not SO(D) where D is space-time dimension. This surprising fact leads us to understand a doubling phenomena that fermionic degrees are 2^<(2D)/2> not 2^<D/2> in a unit lattice. Supersymmetric theories have a balance between bosonic and fermionic degree of freedom, and the counterpart for gauge freedom remains as an open problem.Why is the realization of supersymmetry difficult unlike gauge symmetry? One of reasons is the path integral measure of fermi fields. The simplest case is a 0-dimensional case and we have studied the global anomaly under the existence of Majorana fermions. The consequence is that the consistent theory enforces us even number of Majorana fermions in the theory such as Witten's anomaly. On the other hand, how can we set a Majorana fermion on lattice? According to an approach of Luescher for a chiral symmetry, we have tried a Majorana condition. To our regret, the result is not satisfied with chiral Yukawa couplings. Therefore, we need to get systematical approach of symmetry in regularized filed theories. The quantum master equation can describe the realization necessarily and sufficiently. This approach can apply to a system with external gravitational fields.