Multi-moment maps on nearly K\"ahler six-manifolds

TL;DR

This paper investigates multi-moment maps generated by two-torus actions on six-dimensional nearly K"ahler manifolds, deriving explicit formulas, analyzing orbits, and illustrating the structure with concrete examples.

Contribution

It provides explicit expressions for multi-moment maps and characterizes the orbit structure on homogeneous nearly K"ahler six-manifolds with two-torus symmetry.

Findings

Explicit formulas for multi-moment maps are derived.

The structure of stationary orbits and fixed points is characterized.

A trivalent graph representation of the orbit space is constructed.

Abstract

We study multi-moment maps induced by a two-torus action on the four homogeneous nearly K\"ahler six-manifolds. Their explicit expression and stationary orbits are derived. The configuration of fixed-points and one-dimensional orbits is worked out for generic six-manifolds equipped with an $\mathrm{SU}(3)$-structure admitting a two-torus symmetry. Projecting the subspaces obtained to the orbit space yields a trivalent graph. We illustrate this result concretely on the homogeneous nearly K\"ahler examples.