The Overlap-Dirac Operator: Topology and Chiral Symmetry Breaking

We review the spectral flow techniques for computing the index of the overlap Dirac operator including results relevant for SUSY Yang-Mills theories. We describe properties of the overlap Dirac operator, and methods to implement it numerically. We use the results from the spectral flow to illuminate the difficulties in numerical calculations involving domain wall and overlap fermions. PACS. 11.15.Ha- Lattice gauge theory. PACS. 12.38.Gc- Lattice QCD calculations. I. Overlap and domain wall Dirac operators In these proceedings, we review some basic properties of the overlap Dirac operator and how its index can be computed by spectral flow techniques. One of the side results is that for fermions in the adjoint representation of SU(N) we find evidence for fractional topological charge. The presentation is pedagogical with the intent of illustrating the origin of numerical difficulties in simulating overlap and domain wall fermions. Recent results from our work using overlap fermions can be found in references [1, 2, 3, 4]. The massive overlap Dirac operator derived from the overlap formalism [5] is Dov(µ) = 1 2 [1 + µ+ (1 − µ)γ5(HL(m))] (1) where HL(m) is a lattice hermitian Dirac-like operator describing a single fermion species with a large negative mass. The mass m is a regulator parameter for the theory. In this work, we use the hermitian Wilson-Dirac operator Hw(m) = γ5DWilson(−m), although we have tested other fermion actions. The mass parameter −1 < µ < 1 is related to the fermion mass by [6] mf = Z−1m µ(1 +O(a2)). (2) The propagator for external fermions is given by D̃−1(µ) = (1 − µ)−