Lattice QCD with fixed topology

The overlap Dirac operator, which satisfies the Ginsparg-Wilson relation, realizes exact chiral symmetry on the lattice. It also avoids fermion doubling but its locality and smoothness are subtle. In fact, the index theorem on the lattice implies that there are certain points where the overlap Dirac operator has discontinuities. Aside from the theoretical subtleties, this non-smoothness also raises practical problems in numerical simulations, especially in Nf 6= 0 full QCD case. One must carefully calculate the lowest eigenvalue of the Dirac operator at each hybrid Monte Carlo step, in order to catch sudden jumps of the fermion determinant on the topology boundaries (reflection/refraction). The approximation of the sign function, which is a crucial point in implementing the overlap Dirac operator, gets worse near the discontinuities. A solution may be to concentrate on a fixed topological sector in the full theory. It is known that an “admissibility” condition, which suppress small plaquette values, preserves topology of the gauge fields and improves the locality of the overlap Dirac operator at the same time. In this thesis, we test a gauge action which automatically generates “admissible” configurations, as well as (large) negative mass Wilson fermion action which would also keep the topology. The quark potential and the topology stability are investigated with different lattice sizes and different couplings. Then we discuss the effects of these new approaches on the numerical cost of the overlap fermions. The results of quenched QCD in the ǫ-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.