Abstract
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.
Highlights
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
A topologicallytrivial 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
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, Eq (30)
Summary
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. If one attempts to use the lattice [3] to regularize the theory nonperturbatively, not every Ginsparg-Wilson lattice Dirac operator [4] might possess topological zero modes1 with index satisfying (1), even though it is exponentially-local, doublers-free, and reproduces correct axial anomaly for topologically-trivial gauge backgrounds. 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
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