# Hypersymplectic manifolds and associated geometries

## TL;DR

This paper explores the geometric structure of hypersymplectic manifolds with SU(1,1) symmetry, identifying conditions under which they form cones over split 3-Sasakian manifolds and fibrations over para-quaternionic Kähler manifolds, with applications to Nahm-Schmid equations.

## Contribution

It introduces an obstruction criterion for hypersymplectic manifolds with SU(1,1) actions and characterizes their geometric structures when the obstruction vanishes, including new examples.

## Key findings

- Hypersymplectic manifolds with vanishing obstruction are metric cones over split 3-Sasakian manifolds.
- Proper SU(1,1) actions induce fibrations over para-quaternionic Kähler manifolds.
- The moduli space of Nahm-Schmid equations admits a fibration over a para-quaternionic Kähler manifold.

## Abstract

We investigate an obstruction for hypersymplectic manifolds equipped with a free, isometric action of SU(1,1). When the obstruction vanishes, we show that the manifold is a metric cone over a split 3-Sasakian manifold. Furthermore, if the action of SU(1,1) is also proper, then the hypersymplectic manifold fibres over a para-quaternionic Kahler manifold. We conclude the article with some examples for which the obstruction vanishes. In particular, we show that the moduli space to Nahm-Schmid equations admits a fibration over a para-quaternionic Kahler manifold.