Gauge theory and G2-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-closed ${\mathrm{G}}_2$-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.

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Cited by in corpus (6)

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• Gauge theory on Aloff-Wallach spaces
• Instantons on Sasakian 7-manifolds
• Isolated singularities in $G_2$-structures with torsion