# Instantons on hyperk\"ahler manifolds

## TL;DR

This paper explores instantons on hyperk"ahler manifolds, establishing a local correspondence with holomorphic functions and providing a streamlined proof of Uhlenbeck's Compactness Theorem in this context.

## Contribution

It introduces a new local bijection between instantons and holomorphic functions, simplifying the proof of a key compactness theorem for hyperk"ahler manifolds.

## Key findings

- Established a local bijection between instantons and holomorphic functions.
- Provided a streamlined proof of Uhlenbeck's Compactness Theorem.
- Enhanced understanding of instanton moduli spaces on hyperk"ahler manifolds.

## 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.