Cohomological lifting of multi-toric graphs

We study $G_2$-manifolds obtained from circle bundles over symplectic $SU(3)$-manifolds with $T^2$-symmetry. When the geometry is multi-Hamiltonian, we show how the compact part of the resulting multi-moment graph for the $G_2$-structure may obtained cohomologically from the base. The lifting procedure is illustrated in the context of toric geometry.