Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 017, 23 pages      arXiv:1207.6938

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

Anda Degeratu a and Thomas Walpuski b
a) University of Freiburg, Mathematics Institute, Germany
b) Massachusetts Institute of Technology, Department of Mathematics, USA

Received June 02, 2015, in final form February 06, 2016; Published online February 15, 2016

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.

Key words: crepant resolutions; HYM connections.

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