JournalsrmiVol. 36, No. 6pp. 1753–1778

Gauge theory and G2\mathrm G_2-geometry on Calabi–Yau links

  • Omegar Calvo-Andrade

    Centro de Investigación en Matemáticas, Guanajuato, Mexico
  • Lázaro O. Rodríguez Díaz

    Universidade Federal do Rio de Janeiro, Brazil
  • Henrique N. Sá Earp

    Universidade Estadual de Campinas, Brazil
Gauge theory and $\mathrm G_2$-geometry on Calabi–Yau links cover

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The 7-dimensional link KK of a weighted homogeneous hypersurface on the round 9-sphere in C5\mathbb{C}^5 has a nontrivial null Sasakian structure which is contact Calabi–Yau, in many cases. It admits a canonical co-calibrated G2\mathrm G_2-structure φ\varphi induced by the Calabi–Yau 3-orbifold basic geometry. We distinguish these pairs (K,φ)(K,\varphi) by the Crowley–Nordström Z48\mathbb{Z}_{48}-valued ν\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 G2\mathrm G_2-instantons on holomorphic Sasakian bundles over KK are exactly the transversely Hermitian Yang–Mills connections. As a proof of principle, we obtain G2\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 G2\mathrm G_2-instanton count.

Cite this article

Omegar Calvo-Andrade, Lázaro O. Rodríguez Díaz, Henrique N. Sá Earp, Gauge theory and G2\mathrm G_2-geometry on Calabi–Yau links. Rev. Mat. Iberoam. 36 (2020), no. 6, pp. 1753–1778

DOI 10.4171/RMI/1182