Communications in Analysis and Geometry

Volume 30 (2022)

Number 2

From vortices to instantons on the Euclidean Schwarzschild manifold

Pages: 335 – 380



Ákos Nagy (Department of Mathematics, Duke University. Durham, North Carolina, U.S.A.)

Gonçalo Oliveira (IMPA and Universidade Federal Fluminense IME–GMA, Niterói, Brazil)


The first irreducible solution of the $\mathrm{SU}(2)$ anti-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 ES manifold has resisted many efforts for the past 40 years.

By exploring a correspondence between the planar Abelian vortices and spherically symmetric instantons on the ES manifold, 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 noninteger energy and nontrivial 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.

Received 6 August 2018

Accepted 9 August 2019

Published 29 November 2022