A hyperholomorphic line bundle on certain hyperk\"ahler manifolds not admitting an $S^1$-symmetry

Generalizing work of Haydys and Hitchin, we prove the existence of a hyperholomorphic line bundle on certain hyperk"ahler manifolds that do not necessarily admit an $S^1$ action. As examples, we consider the moduli space of (non-strongly) parabolic Higgs bundles, the moduli space of solutions to Nahm’s equations, and Nakajima quiver varieties.