Closed forms and multi-moment maps

TL;DR

This paper generalizes multi-moment maps to geometries with closed forms of any degree, establishing existence, uniqueness, and examples, especially for degree four forms related to special holonomy.

Contribution

It introduces a broad extension of multi-moment maps, providing fundamental theorems and classifications for geometries with higher-degree closed forms.

Findings

Multi-moment maps exist and are unique for degree four forms under (3,4)-trivial symmetry groups.

Structural descriptions of (3,4)-trivial Lie algebras are provided.

Examples include geometries associated with special holonomy.

Abstract

We extend the notion of multi-moment map to geometries defined by closed forms of arbitrary degree. We give fundamental existence and uniqueness results and discuss a number of essential examples, including geometries related to special holonomy. For forms of degree four, multi-moment maps are guaranteed to exist and are unique when the symmetry group is (3,4)-trivial, meaning that the group is connected and the third and fourth Lie algebra Betti numbers vanish. We give a structural description of some classes of (3,4)-trivial algebras and provide a number of examples.