From vortices to instantons on the Euclidean Schwarzschild manifold

TL;DR

This paper classifies and constructs new instantons on the Euclidean Schwarzschild manifold, revealing solutions with non-integer energy and non-trivial holonomy, and disproving a previous conjecture about their symmetry properties.

Contribution

It provides a complete classification of spherically symmetric SU(2) instantons on the Euclidean Schwarzschild manifold, including new solutions with non-integer energy and non-trivial holonomy.

Findings

Complete description of a moduli space component of instantons.

New examples of instantons with non-integer energy.

Disproof of a conjecture regarding symmetry invariance.

Abstract

The first irreducible solution of the $\mathrm{SU} (2)$ self-duality equations on the Euclidean Schwarzschild (ES) manifold was found by Charap and Duff in 1977, only 2 years later than the famous BPST instantons on $\mathbb{R}^4$ were discovered. While soon after, in 1978, the ADHM construction gave a complete description of the moduli spaces of instantons on $\mathbb{R}^4$, the case of the Euclidean Schwarzschild manifold has resisted many efforts for the past 40 years. By exploring a correspondence between the planar Abelian vortices and spherically symmetric instantons on ES, we obtain: a complete description of a connected component of the moduli space of unit energy $\mathrm{SU} (2)$ instantons; new examples of instantons with non-integer energy (and non-trivial holonomy at infinity); a complete classification of finite energy, spherically symmetric, $\mathrm{SU} (2)$ instantons. As opposed to the previously known solutions, the generic instanton coming from our construction is not invariant under the full isometry group, in particular not static. Hence disproving a conjecture of Tekin.