An Ansatz for Hyperk\"ahler $8$-Manifolds with two Commuting Rotating Killing Fields

TL;DR

This paper introduces a new approach to hyperk"ahler 8-manifolds with specific symmetries, reducing the geometric data to a single function satisfying Monge-Ampere type equations expressed via a Poisson bracket.

Contribution

It provides a novel ansatz that simplifies the structure of certain hyperk"ahler 8-manifolds with commuting Killing fields into a single function governed by Monge-Ampere equations.

Findings

Reduction to a single function H of two complex and two real variables

Derivation of six Monge-Ampere type equations for H

Compact expression using a Poisson bracket

Abstract

We consider a hyperk\"{a}hler $8$-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.