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 $SL_2(\mathbb{C})$-bundle over $M$. Our reformulation affords a streamlined proof of Uhlenbeck's Compactness Theorem for instantons on (pseudo-)hyperk\"ahler manifolds.