A numerical test of the continuum index theorem on the lattice

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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 12⁴ lattice using the standard single plaquette Wilson action at a coupling of beta=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 <Q²>=3.3(4) to be compared with <Q²> = 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.