(Anti-)Instantons and the Atiyah-Hitchin Manifold

TL;DR

This paper explores the Atiyah-Hitchin manifold's role in various physical theories, analyzing instanton corrections and proposing a new interpretation involving bound states of instantons and anti-instantons.

Contribution

It provides a detailed analysis of exponentially small instanton corrections and introduces a novel perspective on semi-classical configurations as Euclidean open branes.

Findings

Instanton and anti-instanton bound states with specific charge differences.

Exponential corrections interpreted as infinite series of instanton effects.

Proposal of Euclidean open branes as relevant semi-classical configurations.

Abstract

The Atiyah-Hitchin manifold arises in many different contexts, ranging from its original occurrence as the moduli space of two SU(2) 't Hooft-Polyakov monopoles in 3+1 dimensions, to supersymmetric backgrounds of string theory. In all these settings, (super)symmetries require the metric to be hyperk\"ahler and have an SO(3) transitive isometry, which in the four-dimensional case essentially selects out the Atiyah-Hitchin manifold as the only such smooth manifold with the correct topology at infinity. In this paper, we analyze the exponentially small corrections to the asymptotic limit, and interpret them as infinite series of instanton corrections in these various settings. Unexpectedly, the relevant configurations turn out to be bound states of $n$ instantons and $\bar n$ anti-instantons, with $|n-\bar n|=0,1$ as required by charge conservation. We propose that the semi-classical configurations relevant for the higher monopole moduli space are Euclidean open branes stretched between the monopoles.