Towards A New Formulation of Unified Model with Gravity on a Lattice

我在这一时期的研究主题可以分为以下4类：1)延续之前的调查。2)格上具有非交换性的费米子公式3)新的超空间公式的提出4)格上的超对称1)在已发现的广义规范理论公式的基础上，应用了具有SU（2|1）规范群的截断版广义Yang-Mills理论，导出了Weinberg-Salam模型。对二维量子表面分形维数的c依赖性进行了数值计算，结果与之前的解析结果完全一致。2)我们可以引入格差分算子的温和非交换性来满足莱布尼茨规则。然后我们可以定义格上的微分形式并引入结合法Clifford积。我们澄清了晶格上具有非交换性的狄拉克-卡勒费米子公式。3)已经认识到，拓扑场论量子化中出现的扭曲可以被识别为狄拉克-卡勒费米子公式。我们发现这种扭转机制是出现在拓扑场论量子化中的一个普遍特征，并且在其背后隐藏着一个扭曲的超空间公式。在二维空间中建立了N=2的扭曲超空间公式。然后将这个公式推广到四维N=4。4)通过引入二维N=2扭曲超空间的温和不可交换性，成功地在晶格上构造了扭曲超空间的公式。现在可以把这个公式应用于晶格上N=2的超杨-米尔斯作用。

My research thema in this period can be classified into the following 4 categories :1)Continuation from the previous investigations.2)Fermion Formulation with Non-commutativity on a Lattice.3)A Proposal of New Superspace Formulation.4)Supersymmetry on a Lattice.1)Based on the generalized gauge theory formulation which we discovered, we applied a truncated version of generalized Yang-Mills theory with SU(2|1) gauge group and derived Weinberg-Salam model. Numerical evaluation of the c-dependence of the fractal dimension of two-dimensional quantum surface was investigated and the result is perfect agreement with the analytic result which we obtained previously.2)We may introduce mild non-commutativity for the lattice difference operator tosatisfy the Leibniz rule. We can then define differential form on a lattice and introduce associative Clifford product. We have clarified the Dirac-Kahler fermion formulation with non-commutativity on a lattice.3)It was already recognized that the twisting appeared in the quantization of topological field theory can be identified as a Dirac-Kahler fermion formulation. We found that this twisting mechanism is a universal characteristic which appears in the quantization of topological field theory and a twisted superspace formulation is hidden behind the formulation. N=2 twisted superspace formulation was formulated in two dimensions. Then this formulation is extended into N=4 in four dimensions.4)Twisted superspace formulation is successfully formulated on a lattice with the introduction of the mild non-commutatibity for N=2 twisted superspace in two dimensions. It is now possible to apply this formulation to N=2 super Yang-Mills action on a lattice.