The index theorem in QCD with a finite cut-off

The fixed point Dirac operator on the lattice has exact chiral zero modes on topologically non-trivial gauge field configurations independently whether these configurations are smooth, or coarse. The relation nL−nR=QFP, where nL (nR) is the number of left (right)-handed zero modes and QFP is the fixed point topological charge holds not only in the continuum limit, but also at finite cut-off values. The fixed point action, which is determined by classical equations, is local, has no doublers and complies with the no-go theorems by being chirally non-symmetric. The index theorem is reproduced exactly, nevertheless. In addition, the fixed point Dirac operator has no small real eigenvalues except those at zero, i.e. there are no `exceptional configurationsʹ.