Instantons, three-dimensional gauge theory, and the Atiyah-Hitchin manifold

Abstract

We investigate quantum effects on the Coulomb branch of three-dimensional N = 4 supersymmetric gauge theory with gauge group SU(2). We calculate perturbative and one-instanton contributions to the Wilsonian effective action using standard weakcoupling methods. Unlike the four-dimensional case, and despite supersymmetry, the contribution of non-zero modes to the instanton measure does not cancel. Our results allow us to fix the weak-coupling boundary conditions for the differential equations which determine the hyper-Kähler metric on the quantum moduli space. We confirm the proposal of Seiberg andWitten that the Coulomb branch is equivalent, as a hyper- Kähler manifold, to the centered moduli space of two BPS monopoles constructed by Atiyah and Hitchin.