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Abstract

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.