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

Abstract

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-calibrated \mathrm 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 \mathrm G_2 -instantons on holomorphic Sasakian bundles over K are exactly the transversely Hermitian Yang–Mills connections. As a proof of principle, we obtain \mathrm 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 \mathrm G_2 -instanton count.