THE INDEX THEOREM IN QCD WITH A FINITE CUTOFF

The fixed point Dirac operator on the lattice has exact chiral zero mo des on topologically non-trivial gauge field configurations independen tly whether these configurations are smooth, or coarse. The relation n (L) - n(R) = Q(FP) where n(L) (n(R)) is che number of left (right)-han ded zero modes and Q(FP) is the fixed point topological charge holds n ot only in the continuum limit, but also at finite cut-off values. The fixed point action. which is determined by classical equations. is lo cal, has no doublers and complies with the no-go theorems by being chi rally non-symmetric. The index theorem is reproduced exactly, neverthe less. In addition, the fixed point Dirac operator has no small real ei genvalues except those at zero, i.e. there are no 'exceptional configu rations' (C) 1998 Elsevier Science B.V. All rights reserved.