Analytic approach to nonperturbative aspects of gauge theories

为了更好地理解规范场理论的非微扰方面，如拓扑结构、构造形式、超对称性和夸克一致性，发展格点规范理论和有效场论等分析方法是很有帮助的。在格点规范理论方法中，我们研究了满足Ginsparg-Wilson关系的格点Dirac算符的手征规范理论，明确了格点规范场(Fujiwara)的拓扑结构。此外，我们发展了一种上同调方法来分类拓扑不变量。这有助于在晶格上构造手征U(1)规范理论。在此基础上，给出了晶格上SU(2)Xu(1)弱规范理论的费米子测度的具体构造。我们还研究了超对称规范理论的格子正则化问题，给出了低维N=(2，2)超对称Yang-Mills理论(Suzuki)的格子表示.周期格子上的阿贝尔规范理论的拓扑结构可以从环面上的阿贝尔规范理论来理解.我们研究了任意偶数维均匀磁背景环面上的Dirac算符零模(Fujiwara)。波函数的扭曲边界条件可以利用环面的几何性质来求解。本征值问题是完全可以解决的。我们发现出现了一组具有手性的零模，并且零模的简并可以从背景规范场的磁平移对称性中得到理解。我们的结果与指数定理完全一致。在有效场论方法中，我们研究了Yang-Mills场的Cho-Faddeev-Niemi分解，并用非线性场变换(Kondo)重新表述了夸克禁闭的机制。

To have better understandings about nonperturbative aspects of guage field theories such as topological structure, constructive formulation, supersymmetry and quark confimenent it is helpful to develop analytical methods such as lattice gauge theory and effective field theory approaches.In lattice gauge theory approach we investigated chiral gauge theories based on lattice Dirac operator satisfying the Ginsparg-Wilson relation and made it clear the topological structure of lattice gauge fields (Fujiwara). Furthermore, we developeded a cohomological method to classify topological invariants. This tourned out to be useful to facilitate construction of chiral U(1) gauge theories on the lattice. Furthermore, we gave a concrete construction of the fermion measure for SU(2)xU(1) electroweak gauge theory on the lattice (Kikukawa). We also investigated lattice regularization of supersymmetric gauge theories and gave a lattice formulation of N=(2, 2) supersymmetric Yang-Mills theory in lower dimensions (Suzuki).The topological structure of abelian gauge theories on a periodic lattice can be understood from abelian gauge theory on a torus. We investigated Dirac operator zero-modes on a torus with a uniform magnetic background in arbitrary even dimensions (Fujiwara). The twisted boundary conditions for the wave functions can be resolved by making use of the geometric nature of the torus. The eigenvalue problem can be solved completely. We found that a set of zero-mode with a desired chirality appear and the degeneracy of the zero-modes can be understood from the magnetic translation symmetry of the background gauge field. Our results are perfectly consistent with the index theorem.In effective field theory approach we investigated Cho-Faddeev-Niemi decomposition of Yang-Mills fields and explained the mechanism of quark confinement by reformulating the theory in terms of the nonlinear field transformations (Kondo).