Instantons, Three-Dimensional Gauge Theory, and the Atiyah-Hitchin Manifold

TL;DR

This paper addresses the exact structure of the quantum Coulomb branch in the 3D $N=4$ $SU(2)$ gauge theory by computing both perturbative one-loop corrections and the leading nonperturbative one-instanton contributions. A key result is that, unlike in four dimensions, 3D instantons exhibit a spectral asymmetry that prevents a complete cancellation between non-zero modes, and this residual effect fixes the boundary conditions for the differential equations governing the hyper-Kähler metric on the moduli space. The authors show that the combined perturbative and one-instanton data select the Atiyah-Hitchin metric as the exact Coulomb-branch metric, thereby verifying Seiberg-Witten's proposal that the quantum moduli space is equivalent to the centered moduli space of two BPS monopoles. This work provides a first-principles field-theoretic derivation of the AH metric in a supersymmetric gauge theory and highlights the crucial role of spectral asymmetry in three-dimensional instanton physics.

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 weak-coupling 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-Kahler metric on the quantum moduli space. We confirm the proposal of Seiberg and Witten that the Coulomb branch is equivalent, as a hyper-Kahler manifold, to the centered moduli space of two BPS monopoles constructed by Atiyah and Hitchin.