(Anti-)Instantons and the Atiyah-Hitchin Manifold

TL;DR

This work analyzes the exponential, non-perturbative corrections to the Atiyah-Hitchin hyperkähler manifold, casting them as instanton–anti-instanton bound-state effects across multiple string and field theory dualities. By recasting the AH metric in modular/form language and tracing the corresponding instanton configurations in brane setups and 3D $ ext{N}=4$ gauge theories, the authors show that the non-perturbative structure is organized by integer powers of $q=e^{-r}$ with precise phase factors, consistent with a bound-state interpretation. The results challenge conventional zero-mode counting by demonstrating how charge-conserving instanton–anti-instanton configurations contribute to four-fermion couplings, with a semiclassical action $S_{ ext{cl}}=(n+ar n)igl(r- frac{1}{2} ext{log}{rac{(2r)^3}{r-2}}igr) + i(n-ar n)oldsymbol{ au}$ and a modular-symmetric coefficient structure. These insights have implications for non-perturbative corrections to hyperkähler moduli spaces in supersymmetric theories and suggest a universal role for brane-based semi-classical objects in shaping exact metrics.

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 hyperkähler 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.