A numerical test of the continuum index theorem on the lattice

Abstract

The overlap formalism of chiral fermions provides a tool to measure the index, Q, of the chiral Dirac operator in a fixed gauge field background on the lattice. This enables a numerical measurement of the probability distribution, p( Q), in Yang-Mills theories. We have obtained an estimate for p( Q) in pure SU(2) gauge theory by measuring Q on 140 independent gauge field configurations generated on a 12 4 lattice using the standard single plaquette Wilson action at a coupling of β = 2.4. This distribution is in good agreement with a recent measurement [8] of the distribution of the topological charge on the same lattice using the same coupling and the same lattice gauge action. In particular we find 〈 Q 2〉 = 3.3(4) to be compared with 〈 Q 2〉 = 3.9(5) found in [8]. The good agreement between the two distributions is an indication that the continuum index theorem can be carried over in a probabilistic sense on to the lattice.