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

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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.