An Ansatz for Hyperkähler $8$-Manifolds with two Commuting Rotating Killing Fields

Abstract

We consider a hyperkähler $8$-manifold admitting either a $U(1)\times \mathbb{R}$, or a $U(1)\times U(1)$ action, where the first factor preserves $g$ and $I$, and acts on ${\omega }_2+i{\omega }_3$ by multiplying it by itself, while the second factor preserves $g$ and acts triholomorphically. Such data can be reduced to a single function $H$ of two complex variables and two real variables satisfying $6$ equations of Monge-Ampere type, which can be compactly written down using a Poisson bracket.

8 pages

References in corpus (1)

• HyperKahler and quaternionic Kahler manifolds with S^1-symmetries