MULTI-INSTANTONS, 3-DIMENSIONAL GAUGE-THEORY, AND THE GAUSS-BONNET-CHERN THEOREM

We calculate multi-instanton effects in a three-dimensional gauge theo ry with N = 8 supersymmetry and gauge group SU(2). The k-instanton con tribution to an eight-fermion correlator is found to be proportional t o the Gauss-Bonnet-Chem integral of the Gaussian curvature over the ce ntered 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) o ver tilde(2) found by Atiyah and Hitchin. In this case the integral is equal to the Euler character of the manifold. More generally the inte gral 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 boun dary term. We conjecture that the boundary terms vanish and evaluate t he multi-instanton contributions using recent results for the cohomolo gy 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 Else vier Science B.V.