Dirac Eigenmodes in an Environment of Dynamical Fermions

Throughout the last decades of lattice QCD topological excitations have played a crucial and sometimes controversial role. It is natural, in particular in view of the purpose of this volume in honor of Adriano Di Giacomo, that they are a topic also here.1 The topological charge of a gauge configuration is a well-defined concept only in the continuum and for differentiable fields. In quantum field theory the gauge configurations are in general non-differentiable. Lattice discretization turns out to be an ideal tool for doing a non-perturbative gauge invariant regularization of QCD; however introducing a smallest distance thus we lose the continuum in such studies. In fact, a constructive definition of the topological charge is only possible when certain smoothness requirements are introduced.2 It is also quite demanding to implement in actual calculations. For that reason various other methods have been addressed, among them ultra-local definitions of the topological charge density 3, together with various modifications.4 When considering the total topological charge of gauge field configurations cooling techniques have been employed in the attempt to smoothen the gauge configuration and get rid of discretization artefacts like dislocations for example. One of the cleanest tools for identifying the topological charge is through the measurement of a costly fermionic observable: eigenmodes of the Dirac operator. The AtiyahSinger index theorem 5 relates the topological charge of a gauge configuration with the difference between the number of left-handed and right-handed zero modes. Unfortunately the discretization in itself complicates this approach. In the continuum zero modes also have definite chirality. Dirac operators on the lattice violate explicitly chiral symmetry, as it is defined in the continuum. Only recently one has rediscovered a relation6 generalizing the continuum chiral symmetry to the lattice by adding a local piece, that vanishes in the continuum. In its simplest form this so-called Ginsparg-Wilson (GW) relation reads {D, D} = DD , (1)