Trihyperkahler reduction and instanton bundles on CP^3

TL;DR

This paper introduces a new geometric framework called trisymplectic reduction, linking hyperkaehler geometry with instanton moduli spaces on CP^3, and proves a longstanding conjecture about their smoothness and dimension.

Contribution

It develops the theory of trisymplectic structures and reductions, and applies it to prove the smoothness and dimension of instanton moduli spaces on CP^3, resolving a 30-year-old conjecture.

Findings

Moduli space of instanton bundles on CP^3 is smooth and connected.

Dimension of the moduli space for rank 2, charge c instantons is 8c-3.

Trisymplectic reduction is compatible with hyperkaehler reduction.

Abstract

A trisymplectic structure on a complex 2n-manifold is a triple of holomorphic symplectic forms such that any linear combination of these forms has constant rank 2n, n or 0, and degenerate forms in $\Omega$ belong to a non-degenerate quadric hypersurface. We show that a trisymplectic manifold is equipped with a holomorphic 3-web and the Chern connection of this 3-web is holomorphic, torsion-free, and preserves the three symplectic forms. We construct a trisymplectic structure on the moduli of regular rational curves in the twistor space of a hyperkaehler manifold, and define a trisymplectic reduction of a trisymplectic manifold, which is a complexified form of a hyperkaehler reduction. We prove that the trisymplectic reduction in the space of regular rational curves on the twistor space of a hyperkaehler manifold M is compatible with the hyperkaehler reduction on M. As an application of these geometric ideas, we consider the ADHM construction of instantons and show that the moduli space of rank r, charge c framed instanton bundles on CP^3 is a smooth, connected, trisymplectic manifold of complex dimension 4rc. In particular, it follows that the moduli space of rank 2, charge c instanton bundles on CP^3 is a smooth complex manifold dimension 8c-3, thus settling a 30-year old conjecture.