Rigid HYM Connections on Tautological Bundles over ALE Crepant Resolutions in Dimension Three

TL;DR

This paper proves that tautological bundles over ALE crepant resolutions of certain Calabi-Yau orbifolds in three dimensions admit rigid Hermitian-Yang-Mills connections, extending known results from dimension two.

Contribution

It establishes the existence of rigid Hermitian-Yang-Mills connections on tautological bundles in three dimensions using derived category methods.

Findings

Existence of rigid Hermitian-Yang-Mills connections on tautological bundles.

Re-derivation of cohomological identities via Atiyah-Patodi-Singer index theorem.

Dimension three analogues of Kronheimer and Nakajima's results.

Abstract

For $G$ a finite subgroup of ${\rm SL}(3,{\mathbb C})$ acting freely on ${\mathbb C}^3{\setminus} \{0\}$ a crepant resolution of the Calabi-Yau orbifold ${\mathbb C}^3\!/G$ always exists and has the geometry of an ALE non-compact manifold. We show that the tautological bundles on these crepant resolutions admit rigid Hermitian-Yang-Mills connections. For this we use analytical information extracted from the derived category McKay correspondence of Bridgeland, King, and Reid [J. Amer. Math. Soc. 14 (2001), 535-554]. As a consequence we rederive multiplicative cohomological identities on the crepant resolution using the Atiyah-Patodi-Singer index theorem. These results are dimension three analogues of Kronheimer and Nakajima's results [Math. Ann. 288 (1990), 263-307] in dimension two.