Special holonomy manifolds with torus symmetry

The focus of this talk will be Calabi-Yau 3-folds and G2 -manifolds. Both types of spaces come with certain geometric data, including a Ricci-flat metric, making it challenging to find explicit examples. A construction by Foscolo-Haskins-Nordstrom provides a way of constructing G2 -manifolds on the total space of a circle bundle over a certain type of Calabi-Yau 3-fold. I will discuss the case where the Calabi-Yau manifold comes equipped with a 3-torus action (partially) preserving the geometric structure. In this instance, we will see that the G2-manifold also admits a 3-torus action and that the orbit space has a natural parameterisation in terms of so-called multi-moment maps. I will also look at the relationship between the combinatorial data of these two types of toric geometries