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算符，研究了格子手征规范理论的拓扑结构及其在量子理论中的作用。任何格子空间的位形都不可能具有非平凡的拓扑结构，因为任意场的位形都可以连续变形为平凡的位形。通过施加一种光滑性条件来限制构型，有可能引入拓扑上的非平凡结构。在规范场的情况下，这种非平凡的拓扑结构可以通过耦合手征格子费米子来证明。在像QCD这样的矢量型无质量规范理论中，我们可以用重叠的狄拉克算符来表示手征对称的格子作用量。在这种情况下，费米子测量不可能是手征对称的，但会产生雅可比形式的手征反常。重叠狄拉克算符在格点规范场的整个组态空间上没有很好的定义，但它是…的更多定义仅用于流畅的配置。这意味着理论定义明确的格子场的组态空间被断开为几个连通分量。数值分析了厄米-威尔逊-狄拉克算符在阿贝尔规范背景下的谱流，结果表明格子-狄拉克算符的指数随规范场拓扑荷的变化而变化，可以认为是光滑的。通过求出费米子测度在手征变换下的雅可比，我们还建立了任意偶数维非阿贝尔规范背景经典连续统极限中的手征反常。光滑性条件也可能引入无关紧要的拓扑结构，如连通分量中的空洞。我们通过耦合手征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作用量。通过这样做，我们证明了非平凡的存在。即使在与量规异常相关的连通组件中也是如此。较少

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