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

TL;DR

This paper proves the existence of hyperholomorphic line bundles on specific hyperk"ahler manifolds lacking an $S^1$ symmetry, expanding the understanding of geometric structures on these spaces.

Contribution

It generalizes previous results by constructing hyperholomorphic line bundles without requiring an $S^1$-action, applicable to various important moduli spaces.

Findings

Existence of hyperholomorphic line bundles on certain hyperk"ahler manifolds

Application to moduli spaces of parabolic Higgs bundles and Nahm's equations

Extension of previous work by Haydys and Hitchin

Abstract

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.