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Instantons on hyperkähler manifolds

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Devchand,  Chandrashekar
Quantum Gravity & Unified Theories, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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1812.06498.pdf
(Preprint), 435KB

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Citation

Devchand, C., Pontecorvo, M., & Spiro, A. (in preparation). Instantons on hyperkähler manifolds.


Cite as: http://hdl.handle.net/21.11116/0000-0002-BB70-D
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.