Nearly K\"ahler six-manifolds with two-torus symmetry

We consider nearly K\"ahler 6-manifolds with effective 2-torus symmetry. The multi-moment map for the $T^2$-action becomes an eigenfunction of the Laplace operator. At regular values, we prove the $T^2$-action is necessarily free on the level sets and determines the geometry of three-dimensional quotients. An inverse construction is given locally producing nearly K\"ahler six-manifolds from three-dimensional data. This is illustrated for structures on the Heisenberg group.