Instantons on hyperkahler manifolds

An instanton (E, D) on a (pseudo-)hyperkahler manifold M is a vector bundle E associated with a principal G-bundle with a connection D whose curvature is pointwise invariant under the quaternionic structures of TxM,x is an element of 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 GC defined on an appropriate SL2(C)-bundle over M. Our reformulation affords a streamlined proof of Uhlenbeck's compactness theorem for instantons on (pseudo-)hyperkahler manifolds.