Noncommutative chiral anomaly and the Dirac-Ginsparg-Wilson operator

It is shown that the local axial anomaly in $2-$dimensions emerges naturallyif one postulates an underlying noncommutative fuzzy structure of spacetime .In particular the Dirac-Ginsparg-Wilson relation on ${\bf S}^2_F$ is shown tocontain an edge effect which corresponds precisely to the ``fuzzy'' $U(1)_A$axial anomaly on the fuzzy sphere . We also derive a novel gauge-covariantexpansion of the quark propagator in the form $\frac{1}{{\calD}_{AF}}=\frac{a\hat{\Gamma}^L}{2}+\frac{1}{{\cal D}_{Aa}}$ where$a=\frac{2}{2l+1}$ is the lattice spacing on ${\bf S}^2_F$, $\hat{\Gamma}^L$ isthe covariant noncommutative chirality and ${\cal D}_{Aa}$ is an effectiveDirac operator which has essentially the same IR spectrum as ${\cal D}_{AF}$but differes from it on the UV modes. Most remarkably is the fact that bothoperators share the same limit and thus the above covariant expansion is notavailable in the continuum theory . The first bit in this expansion$\frac{a\hat{\Gamma}^L}{2}$ although it vanishes as it stands in the continuumlimit, its contribution to the anomaly is exactly the canonical theta term. Thecontribution of the propagator $\frac{1}{{\cal D}_{Aa}}$ is on the other handequal to the toplogical Chern-Simons action which in two dimensions vanishesidentically .Comment: 26 pages, latex fil