Topology and the Dirac operator spectrum in finite-volume gauge theories

The interplay between gauge field winding numbers, θ-vacua, and the Dirac operator spectrum in finite-volume gauge theories is reconsidered. To assess the weight of each topological sector, we compare the mass-dependent chiral condensate in gauge field sectors of fixed topological index with the answer obtained by summing over the topological charge. Also the microscopic Dirac operator spectrum in the full finite-volume Yang-Mills theory is obtained in this way, by summing over all topological sectors with the appropriate weight