Multi-instantons, three-dimensional gauge theory, and the Gauss-Bonnet-Chern theorem

We calculate multi-instanton effects in a three-dimensional gauge theory with N = 8 supersymmetry and gauge group SU(2). The k-instanton contribution to an eight-fermion correlator is found to be proportional to the Gauss-Bonnet-Chem integral of the Gaussian curvature over the centered moduli space of charge-k BPS monopoles, (M) over tilde(k). For k = 2 the integral can be evaluated using the explicit metric on (M) over tilde(2) found by Atiyah and Hitchin. In this case the integral is equal to the Euler character of the manifold. More generally the integral is the volume contribution to the index of the Euler operator on (M) over tilde(k), which may differ from the Euler character by a boundary term. We conjecture that the boundary terms vanish and evaluate the multi-instanton contributions using recent results for the cohomology of (M) over tilde(k). We comment briefly on the implications of our result for a recently proposed test of M(atrix) theory. (C) 1997 Elsevier Science B.V.