Instantons on hyperk\"ahler manifolds

An instanton $(E,D)$ on a (pseudo-)hyperk"ahler manifold $M$ is a vector bundle $E$ associated to a principal $G$-bundle with a connection $D$ whose curvature is pointwise invariant under the quaternionic structures of $T_x M,
x\in M$, and thus satisfies the Yang-Mills equations. Revisiting a construction of solutions, we prove a local bijection between gauge equivalence classes of instantons on $M$ and equivalence classes of certain holomorphic functions taking values in the Lie algebra of $G^{\mathbb{C}}$ defined on an appropriate $S{L}_2(\mathbb{C})$-bundle over $M$. Our reformulation affords a streamlined proof of Uhlenbeck’s Compactness Theorem for instantons on (pseudo-)hyperk"ahler manifolds.