The index theorem in QCD with a finite cut-off

TL;DR

The paper proves that the fixed-point (FP) lattice formulation of QCD, with the FP Dirac operator $h(U)$ and FP topological charge $Q^{FP}$, realizes the lattice index theorem at finite cut-off via $n_L - n_R = Q^{FP}$, independent of gauge-field smoothness. The FP action is local, has no doublers, and satisfies $h^ abla = abla$ and $h^\

Abstract

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 $n_L-n_R = Q^{FP}$, where $n_L$ $(n_R)$ is the number of left (right)-handed zero modes and $Q^{FP}$ 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 configurations'.

Figures (1)

• Figure 1: The spectrum of the fixed point Dirac operator: a) for the special case $R_{nn'}=1/\kappa_{\rm f}\cdot\delta_{nn'}$ the eigenvalues lie on a circle, b) for a more general set of block transformations the eigenvalues lie between the two circles.