Lattice QCD with fixed topology

The overlap Dirac operator, which satisfies the Ginsparg-Wilson relation, realizes exact chiral symmetry on the lattice without any unphysical doubler modes. To perform the path integrals, one should, however, note that the overlap fermion determinant has discontinuities where the topology is ill-defined, its locality is doubtful, and the numerical cost is suddenly enhanced. An interesting solution would be to concentrate on a fixed topological sector in the full configuration space. In this thesis, we test a gauge action which automatically satisfies Luescher's admissibility condition, as well as an additional (large) negative mass Wilson fermion action in the quenched study (with no light quarks). Both of them would keep the topology along the HMC simulations. The quark potential and the topology stability are investigated with different lattice sizes and different couplings. The results of quenched QCD in the $\epsilon$-regime are also presented as an example of the lattice studies with fixed topology. Remarkable quark mass and topology dependences of meson correlators allow us to determine the fundamental parameters of the effective theory, in which the exact chiral symmetry with the Ginsparg Wilson relation plays a crucial role