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

Abstract

For $G$ a finite subgroup of $\mathrm{SL}(3,\mathbb{C})$ 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.

References in corpus (3)

• Derived categories for the working mathematician
• Fredholm theory for elliptic operators on quasi-asymptotically conical spaces
• Partial Resolutions of Orbifold Singularities via Moduli Spaces of HYM-type Bundles

Cited by in corpus (3)

• The Kähler Quotient Resolution of ${\mathbb{C}}^3/\Gamma$ singularities, the McKay correspondence and D=3 $\mathcal{N}=2$ Chern-Simons gauge theories
• Resolution a la Kronheimer of ${\mathbb{C}}^3/\Gamma$ singularities and the Monge-Ampere equation for Ricci-flat Kaehler metrics in view of D3-brane solutions of supergravity
• Crepant Resolutions of ${\mathbb{C}}^3/{\mathbb{Z}}_4$ and the Generalized Kronheimer Construction (in view of the Gauge/Gravity Correspondence)