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

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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