Quaternionic toric manifolds

TL;DR

This paper introduces quaternionic toric manifolds, extending toric geometry into the quaternionic setting, and explores their structures, convexity properties, and potential as a testing ground for quaternionic geometric theories.

Contribution

It develops a construction of quaternionic toric manifolds from Delzant polytopes, establishing their geometric structures and properties.

Findings

Manifolds are of real dimension 4m with Sp(1) actions.

Existence of a 4-plectic structure and generalized moment map.

Convexity properties of the moment map image studied.

Abstract

In the present paper we introduce and study a new notion of toric manifold in the quaternionic setting. We develop a construction with which, starting from appropriate $m$-dimensional Delzant polytopes, we obtain manifolds of real dimension $4m$, acted on by $m$ copies of the group ${\rm Sp}(1)$ of unit quaternions. These manifolds are quaternionic regular and can be endowed with a $4$-plectic structure and a generalized moment map. Convexity properties of the image of the moment map are studied. Quaternionic toric manifolds appear to be a large enough class of examples where one can test and study new results in quaternionic geometry.