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 $\phi$ induced by the Calabi–Yau 3-orbifold basic geometry. We distinguish these pairs $(K,\phi )$ 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.