Let \((N^n,g)\) be a Riemannian manifold with holonomy group \(H\). A connection 1-form \(A\) on a principal \(G\)-bundle over \(N^n\) is called an instanton if its curvature form \(F_A:=dA+A\wedge A\) lies in the subbundle \(\mathfrak{h}\subset so(n)\cong\Lambda^2(N)\). If instead \(F_A\) belongs to the complement, that is \(F_A\in\mathfrak{h}^\perp\), then \(A\) is called an anti-instanton.
The author studies the existence of abelian (anti-)instantons on quaternionic Kähler 8-manifolds \(N^8\), that is \(H\subseteq\mathrm{Sp}(2)\mathrm{Sp}(1)\) but \(H\not\subset\mathrm{Sp}(2)\). Here abelian means that the structure group of the principal \(G\)-bundle is \(G=\mathrm{U}(1)\).
Y. S. Poon and S. M. Salamon showed in [J. Differ. Geom. 33, No. 2, 363–378 (1991; Zbl 0733.53035)] that the following are the only (complete) quaternionic Kähler 8-manifolds of positive scalar curvature (these are necessarily compact): \[\mathbb{HP}^2=\frac{\mathrm{Sp}(3)}{\mathrm{Sp}(2)\mathrm{Sp}(1)},\qquad\mathrm{Gr}_2(\mathbb{C}^4)=\frac{\mathrm{SU}(4)}{\mathrm{S}(\mathrm{U}(2)\mathrm{U}(2))},\qquad\mathrm{Gr}_2(\mathbb{C}^4)= \frac{\mathrm{G}_2}{\mathrm{SO}(4)}.\]
The author shows in Corollary 2.9 that no abelian anti-instantons exist on compact quaternionic Kähler 8-manifolds of positive scalar curvature.
In the noncompact setting, the author constructs explicit examples of abelian anti-instantons on the (complete) quaternionic Kähler 8-manifolds of negative scalar curvature obtained by the author in [Commun. Math. Phys. 403, No. 1, 1–35 (2023; Zbl 1528.53051)], see Theorem 2.5 and Section 3.