From vortices to instantons on the Euclidean Schwarzschild manifold (Q2107253)

The paper investigates a connection between spherically symetric SU\((2)\) instantons and planar abelian vortices. In the setting of the Euclidean Schwartzschild (ES) manifold, the authors then obtain the following: \begin{itemize} \item a description of a connected component of the moduli space of unit energy SU\((2)\) instantons. \item new examples of instantons with noninteger energy and nontrivial holonomy at infinity, \item as well as a complete classification of finite energy, spherically symmetric, SU\((2\)) instanton. In particular, Uhlenbeck compactness is investigated for these instantons \end{itemize} Recall that instantons are anti-self dual connections with finite Yang-Mills energy \[ E_{YM}=\frac{1}{8\pi^2}\int_M |\mathcal{F}_D|^2\mathrm{vol}_g. \] Here \(\mathcal{D}\) is a connection on a principal \(G\)-bundle, for \(G\) a compact semisimple Lie group over an oriented (\(4\)-dimensional) Riemannian manifold \((M,g)\) with associated curvature form \(\mathcal{F}_D\). The connection \(D\) is called anti-self dual if the part of \(\mathcal{F}_D\) lying in the positive eigenspace of the Hodge star operator vanishes. As mentioned, the authors study instantons on the ES manifold, i.e. a certain Ricci-flat, noncompact, but complete manifold which is not Hyper-Kähler. One of the consequences of the present work on the ES manifold is to disprove a conjecture due to \textit{B. Tekin} [Phys Rev. D 65, No. 8, Article ID 084035, 5 p. (1982; \url{doi:10.1103/PhysRevD.65.084035})].