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.
References
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- Weimin Chen and Yongbin Ruan, Orbifold Gromov-Witten theory, Orbifolds in mathematics and physics (Madison, WI, 2001) Contemp. Math., vol. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 25–85. MR 1950941, DOI https://doi.org/10.1090/conm/310/05398
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References
- Jim Bryan and Tom Graber. The crepant resolution conjecture. arXiv: math. AG/ 0610129.
- Jim Bryan, Sheldon Katz, and Naichung Conan Leung. Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds. J. Algebraic Geom., 10(3):549–568, 2001. Preprint version: math.AG/9911056. MR 1832332 (2002j:14047)
- Weimin Chen and Yongbin Ruan. Orbifold Gromov-Witten theory. In Orbifolds in mathematics and physics (Madison, WI, 2001), volume 310 of Contemp. Math., pages 25–85. Amer. Math. Soc., Providence, RI, 2002. MR 1950941 (2004k:53145)
- Hélène Esnault and Eckart Viehweg. Logarithmic de Rham complexes and vanishing theorems. Invent. Math., 86(1):161–194, 1986. MR 853449 (87j:32088)
- C. Faber and R. Pandharipande. Logarithmic series and Hodge integrals in the tautological ring. Michigan Math. J., 48:215–252, 2000. With an appendix by Don Zagier, Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786488 (2002e:14041)
- Phillip Griffiths, editor. Topics in transcendental algebraic geometry, volume 106 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1984. MR 756842 (86b:14004)
- Sheldon Katz. Small resolutions of Gorenstein threefold singularities. In Algebraic geometry: Sundance 1988, pages 61–70. Amer. Math. Soc., Providence, RI, 1991. MR 1108632 (92f:14001)
- David Mumford. Towards an enumerative geometry of the moduli space of curves. In Arithmetic and geometry, Vol. II, pages 271–328. Birkhäuser Boston, Boston, Mass., 1983. MR 717614 (85j:14046)
- Cumrun Vafa. String vacua and orbifoldized LG models. Modern Phys. Lett. A, 4(12):1169–1185, 1989. MR 1016963 (91g:81128)
- Eric Zaslow. Topological orbifold models and quantum cohomology rings. Comm. Math. Phys., 156(2):301–331, 1993. MR 1233848 (94i:32045)
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