The Overlap Formalism and Topological Susceptibility on the Lattice ∗

Two recent methods[1,2] for measuring the topological susceptibility, χt = 〈Q〉 V , in pure SU(2) gauge theory using the lattice regularization are considered to be in disagreement. In one method[1], the gauge fields are smoothed with an improved cooling technique while the topological charge Q is calculated using a lattice discretization. In the other method[2], inverse-blocking is employed to smooth the lattice configurations, after which the topological charge is calculated using an “algebraic” definition. In both methods, scaling is assumed and a continuum limit for χt obtained, with the value of Ref. [1] being about 20% smaller than that of Ref. [2]. It should be noted that the standard Wilson plaquette action is used in the first method to generate the gauge field configurations on the lattice, while in the second method the gauge field ensemble is generated with a certain fixed-point action. Therefore, in addition to scaling, it is also necessary to assume universality to compare the two numbers above. The index of the Euclidean chiral Dirac operator on the lattice may be extracted by the overlap method[3] in a clean manner, that is to say, in a way that does not require the smoothing of background gauge fields. As the Atiyah-Singer index theorem[4] equates the index to the topological charge, the overlap method should thus be useful in probing for topological structure. One