Clover improvement , spectrum and Atiyah - Singer index theorem for the Dirac operator on the lattice ∗

We study the role of the O(a)-improving clover term for the spectrum of the lattice Dirac operator using cooled and thermalized SU(2) gauge field configurations. For cooled configurations we observe improvement of the spectral properties when adding the clover term. For the thermalized case (12 4 , β = 2.4) without clover term we find a rather bad separation of physical and doubler branches making a probabilistic interpretation of the Atiyah-Singer index theorem on the lattice questionable for this β and lattice size. Adding the clover term leads to the creation of additional real eigenvalues which come in pairs of opposite chirality thus further worsening the situation for the index theorem. 1. Introduction In the last few years the spectrum of the Dirac operator on the lattice has seen a lot of attention. This is partly motivated by the hope that a lattice-regularized gauge theory could put the idea of decomposing the fully quan-tized path integral into topological sectors on a conceptually sound basis. In the continuum gauge fields can be classified with respect to their topo-logical charge when they are smooth and obey certain boundary conditions. On the other hand the fields which carry the measure in a continuum path integral do not obey these conditions. The topological arguments thus can only be implemented on a semi-classical level. Lattice regularization might provide the framework to set up the topological concepts in a fully quan-tized theory. Up to so-called exceptional configurations lattice gauge fields can be assigned a topological charge and the exceptional configurations are expected to die out in the continuum limit. A complete decomposition of the path integral into topological sectors could become possible. In the continuum the Atiyah-Singer index theorem [1] relates the topo-logical charge of classical gauge field configurations to the numbers of left-and right-handed zero modes of the Dirac operator. This gives rise to semi-classical results for fermionic observables. Again, these results can not be implemented in the fully quantized model since the gauge fields in the path integral are too rough. On the lattice there is no analytic result for the Atiyah-Singer index theorem, but it has been conjectured [2]-[17] that on the lattice, at least for sufficiently smooth configurations, it is realized in a probabilistic sense. The question however arises if for the parameters where current simulations are done the gauge fields are sufficiently smooth so that the spectrum of the Dirac operator …