Gauge theory and $\mathrm G_2$-geometry on Calabi–Yau links

The 7-dimensional link K of a weighted homogeneous hypersurface on the round 9-sphere in C5 has a nontrivial null Sasakian structure which is contact Calabi–Yau, in many cases. It admits a canonical co-calibrated G2-structure φ induced by the Calabi–Yau 3-orbifold basic geometry. We distinguish these pairs (K,φ) by the Crowley–Nordstrom Z48-valued ν 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 G2-instantons on holomorphic Sasakian bundles over K are exactly the transversely Hermitian Yang–Mills connections. As a proof of principle, we obtain G2-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 G2-instanton count.