The orbifold quantum cohomology of $\mathbb {C}^{2}/\mathbb {Z}_3$ and Hurwitz-Hodge integrals
Authors:
Jim Bryan, Tom Graber and Rahul Pandharipande
Journal:
J. Algebraic Geom. 17 (2008), 1-28
DOI:
https://doi.org/10.1090/S1056-3911-07-00467-5
Published electronically:
July 9, 2007
MathSciNet review:
2357679
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Abstract | References | Additional Information
Abstract: Let $\mathbb {Z}_3$ act on $\mathbb {C}^2$ by non-trivial opposite characters. Let $\mathcal {X}=[\mathbb {C}^{2}/\mathbb {Z}_3]$ be the orbifold quotient, and let $Y$ be the unique crepant resolution. We show that the equivariant genus 0 Gromov-Witten potentials $F^{\mathcal {X}}$ and $F^{Y}$ are equal after a change of variables—verifying the Crepant Resolution Conjecture for the pair $(\mathcal {X},Y)$. Our computations involve Hodge integrals on trigonal Hurwitz spaces, which are of independent interest. In a self-contained Appendix, we derive closed formulas for these Hurwitz-Hodge integrals.
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Additional Information
Jim Bryan
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z4 Canada
ORCID:
0000-0003-2541-5678
Email:
jbryan@math.ubc.ca
Tom Graber
Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, California 91125
Email:
graber@caltech.edu
Rahul Pandharipande
Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000
MR Author ID:
357813
Email:
rahulp@math.princeton.edu
Received by editor(s):
November 30, 2005
Published electronically:
July 9, 2007