Multi-moment maps

TL;DR

This paper introduces multi-moment maps for Lie group actions preserving a closed three-form, establishing their existence under various conditions and illustrating their applications in geometry and holonomy theory.

Contribution

It defines multi-moment maps for a broad class of Lie group actions and characterizes their existence and structure, expanding the theory of moment maps in differential geometry.

Findings

Multi-moment maps exist under mild topological conditions.

All groups with zero second and third Lie algebra Betti numbers admit such maps.

Applications include describing manifolds with G_2 holonomy and torus symmetry.

Abstract

We introduce a notion of moment map adapted to actions of Lie groups that preserve a closed three-form. We show existence of our multi-moment maps in many circumstances, including mild topological assumptions on the underlying manifold. Such maps are also shown to exist for all groups whose second and third Lie algebra Betti numbers are zero. We show that these form a special class of solvable Lie groups and provide a structural characterisation. We provide many examples of multi-moment maps for different geometries and use them to describe manifolds with holonomy contained in G_2 preserved by a two-torus symmetry in terms of tri-symplectic geometry of four-manifolds.