The $7$--dimensional link $K$ of a weighted homogeneous hypersurface on the round $9$--sphere in $\mathbb{C}^5$ has a nontrivial null Sasakian structure which is contact Calabi-Yau, in many cases. It admits a canonical co-closed $\rm G_2$--structure $\varphi$ induced by the Calabi-Yau $3$--orbifold basic geometry. We distinguish these pairs $(K,\varphi)$ by the Crowley-Nordström $\mathbb{Z}_{48}$--valued $\nu$ invariant, for which we prove odd parity and provide an algorithmic formula.

We describe moreover a natural Yang-Mills theory on such spaces, with many important features of the torsion-free case, such as a Chern-Simons formalism and topological energy bounds. In fact, compatible $\rm G_2$--instantons on holomorphic Sasakian bundles over $K$ are exactly the transversely Hermitian Yang-Mills connections. As a proof of principle, we obtain $\rm G_2$--instantons over the Fermat quintic link from stable bundles over the smooth projective Fermat quintic, thus relating in a concrete example the Donaldson-Thomas theory of the quintic threefold with a conjectural $\rm G_2$--instanton count.

This is joint work with Omegar Calvo-Andrade and Lazaro Rodriguez.