Investigations of anomalies and the undex theorem in lattice chiral gauge theories based on noncommutative differential geomerty

基于满足Ginsparg-Wilson关系的Dirac算子，研究了点阵手性规范理论的拓扑结构及其在量子理论中的作用。任何点阵场的组态空间都不可能具有非平凡的拓扑结构，因为任何场的组态都可以连续地变形为平凡的组态。通过施加一种平滑条件来限制构型，可以引入拓扑非平凡结构。在规范场的情况下，这种非平凡拓扑结构的效应可以通过耦合手性晶格费米子来证明。在像QCD这样的矢量型无质量规范理论中，我们可以用重叠狄拉克算子来表述手性对称晶格作用。在这种情况下，费米子测度不能是手性对称的，而是产生雅可比矩阵形式的手性异常。重叠狄拉克算子在晶格规范场的整个构形空间上不是很好定义的，而是只有在光滑构形下才有更多的定义。这意味着理论定义良好的点阵场的构形空间被断开为几个相连的分量。数值分析了阿贝尔规范背景下厄米Wilson-Dirac算子的谱流，证明了晶格Dirac算子的指标随规范场拓扑电荷的变化而变化，可以认为是光滑的。通过计算费米子测度在手性变换下的雅可比矩阵，我们还建立了任意偶数维非阿贝尔规范背景的经典连续统极限中的手性异常。平滑条件还可以引入不平凡的拓扑结构，如连接组件中的孔。我们通过耦合手性费米子研究了这种拓扑结构的影响，并成功地构造了Wess-Zumino-Witten作用。这样我们就建立了非平凡的存在性。即使在连接的部件中，拓扑结构也与量规异常有关。少

Based on the Dirac operator satisfying the Ginsparg-Wilson relation, we have investigated the topological structures of lattice chiral gauge theories and their roles in quantum theory The configuration space of any lattice fields cannot have nontrivial topological structure since any field configurations can be continuously deformed into the trivial one. By restricting the configurations by imposing a kind of smoothness condition it is possible to introduce topologically nontrivial structures. In the case of gauge fields the effects of such nontrivial topological structures can be proved by coupling chiral lattice fermions.In vector-type massless gauge theories like QCD we can formulate chirally symmetric lattice action in terms of the overlap Dirac operator. In this case the fermion measure cannot be chirally symmetric but yields chiral anomaly in the form of Jacobian. The overlap Dirac operator is not well-defined over the entire configuration space of the lattice gauge fields but is … More defined only for smooth configurations. This implies that the configuration space of the lattice fields where the theory is well-defined is disconnected into several connected components. We have numerically analyzed the spectral flows of hermitian Wilson-Dirac operator for abelian gauge backgrounds and shown that the index of the lattice Dirac operator changes in accord with the change of the topological charge for gauge field that can be considered smooth. We have also established the chiral anomaly in the classical continuum limit for the nonabelian gauge backgrounds in arbitrary even dimensions by evaluating the Jacobian of the fermion measure under the chiral transformations.The smoothness condition may also introduces notrivial topological structure like a hole in a connected component. We have investigated the effects of such topological structure by coupling chiral fermions and have succeeded in constructing the Wess-Zumino-Witten action. By doing this we have established the existence of nontrivial. topological structure even in a connected component that is related with the gauge anomalies. Less