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

• Main Authors: Degeratu, Anda (Author), Walpuski, Thomas (Contributor)
• Other Authors: Massachusetts Institute of Technology. Department of Mathematics (Contributor)
• Format: Article
• Language: English
• Published: National Academy of Sciences of Ukraine (SIGMA (Symmetry, Integrability, and Geometry: Methods and Application)), 2017-05-31T19:09:37Z.
• Subjects: Article
• Online Access: Get fulltext

For G a finite subgroup of SL(3,C) acting freely on C[superscript 3]\{0} a crepant resolution of the Calabi-Yau orbifold C[superscript 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.