Numerical study of the lattice index theorem using improved cooling and overlap fermions

We investigate the topological charge and the index theorem on finite lattices numerically. Using mean field improved gauge-field configurations we calculate the topological charge Q using the gluon field definition with $\mathcal{O}{(a}^4)$-improved cooling and an $\mathcal{O}{(a}^4)$-improved field strength tensor $F_{\mu \nu }.$ We also calculate the index of the massless overlap fermion operator by directly measuring the differences between the numbers of zero modes with left- and right–handed chiralities. For sufficiently smooth field configurations we find that the gluon field definition of the topological charge is an integer to better than 1%, and furthermore that this agrees with the index of the overlap Dirac operator, i.e., the Atiyah-Singer index theorem is satisfied. This establishes a benchmark for reliability when calculating lattice quantities that are very sensitive to topology.