The Index of a Ginsparg-Wilson Dirac Operator

A novel feature of a Ginsparg-Wilson lattice Dirac operator is discussed. Unlike the Dirac operator for massless fermions in the continuum, this lattice Dirac operator does not possess topological zero modes for any topologically-nontrivial background gauge fields, even though it is exponentially-local, doublers-free, and reproduces correct axial anomaly for topologically-trivial gauge configurations. PACS numbers: 11.15.Ha, 11.30.Rd, 11.30.Fs In the continuum, the Dirac operator γμ(∂μ + iAμ) of massless fermions in a smooth background gauge field with non-zero topological charge Q has zero eigenvalues and the corresponding eigenfunctions are chiral. The Atiyah-Singer index theorem [1, 2] asserts that the difference of the number of left-handed and right-handed zero modes is equal to the topological charge of the gauge field configuration : n− − n+ = Q . (1) However, if one attempts to use the lattice [3] to regularize the theory nonperturbatively, then not every Ginsparg-Wilson lattice Dirac operator [4] might possess topological zero modes with index satisfying (1), even though it is local, doublers-free, and reproduces correct axial anomaly for topologicallytrivial gauge backgrounds. As a consequence, a topologically-trivial lattice Dirac operator might not realize ’t Hooft’s solution to the U(1) problem in QCD, nor other quantities pertaining to the nontrivial gauge sectors. Nevertheless, from a theoretical viewpoint, it is interesting to realize that one may have the option to turn off the topological zero modes of a GinspargWilson lattice Dirac operator, without affecting its correct behaviors ( axial anomaly, fermion propagator, etc. ) in the topologically-trivial gauge sector. In this paper, I construct an example of such Ginsparg-Wilson lattice Dirac operators, and argue that it does not possess topological zero modes for any topologically-nontrivial gauge configurations satisfying a very mild condition (30). Consider the lattice Dirac operator D = Dc(1I + aDc) −1 (2) with Dc = ∑ μ γμTμ , Tμ = ftμf , (3)