Crepant Resolution Conjecture For Surface Singularities

The classical McKay correspondence is a relation between the representation the-ory of the finite subgroups G of S L(2,C) or S L(3,C) and the cohomology group of thecrepant resolution ofC2/G orC3/G. This correspondence can be interpreted as a corre-spondence between the Chen-Ruan cohomology groups of the orbifolds and the ordinarycohomology groups of the crepant resolutions. Inspired by the classical McKay corre-spondence and some physicists’ works in string theory, Yongbin Ruan, J. Bryan, T. Graberetc. made some conjectures on the correspondence between quantum cohomologies. Ingeneral, for a Gorenstein orbifold, if the crepant resolution exists, the correspondence be-tween Gromov-Witten invariants in arbitrary genera is also called the crepant resolutionconjecture. The most fundamental cases of this conjecture are the ADE surface singu-larities. For surface singularities of type A, part of the conjecture has been solved by T.Coates, A. Corti, H. Iritani, Hsian-Hua Tseng, D. Maulik, and Jian Zhou.For the crepant resolution for surface singularities, the main difculty lies in thecomputation of Hurwitz-Hodge integrals. In this thesis, for nonabelian groupsD n, by J.Bryan and A. Gholampour’s WDVV reduction and normal subgroup reduction, we re-duce the identities of the Hurwitz-Hodge integrals that need to be proved to a small classones. For the latter integrals, we prove a (Laurent) polynomiality. Combining the theseresults and the validity of the quantum McKay correspondence for type A groups, weprove the quantum McKay correspondence type D binary polyhedral groups. We alsostudy the crepant resolution conjecture for total ancestor potentials for surface singulari-ties, and reduce the conjecture to the quantum McKay correspondence in genus zero anda vanishing conjecture for Hurwitz-Hodge integrals. In particular, for singularities of typeA, we prove the conjecture. For the most simple singularity, the A1singularity, we give anew proof for the quantum McKay correspondence, based on the results of A. Bayer andC. Cadman. Finally we propose some related problems that can be studied in the future.