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

TL;DR

This paper studies quantum effects in a 3D supersymmetric gauge theory, calculating instanton contributions to understand the hyper-Kahler metric of the Coulomb branch, confirming a geometric equivalence proposed by Seiberg and Witten.

Contribution

It provides the first calculation of non-zero mode contributions in 3D N=4 gauge theory and confirms the Coulomb branch's hyper-Kahler structure matches the Atiyah-Hitchin manifold.

Findings

Instanton effects modify the Coulomb branch metric.

Non-zero modes do not cancel in 3D despite supersymmetry.

The Coulomb branch is equivalent to 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 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.