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Transactions of the American Mathematical Society

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Localization and gluing of orbifold amplitudes: The Gromov-Witten orbifold vertex

Author: Dustin Ross
Journal: Trans. Amer. Math. Soc. 366 (2014), 1587-1620
MSC (2010): Primary 14N35; Secondary 05A15
Published electronically: November 4, 2013
MathSciNet review: 3145743
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Abstract: We define a formalism for computing open orbifold GW invariants of $ [\mathbb{C}^3/G]$, where $ G$ is any finite abelian group. We prove that this formalism and a suitable gluing algorithm can be used to compute GW invariants in all genera of any toric CY orbifold of dimension $ 3$. We conjecture a correspondence with the DT orbifold vertex of Bryan-Cadman-Young.

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  • [AKMV05] Mina Aganagic, Albrecht Klemm, Marcos Mariño, and Cumrun Vafa, The topological vertex, Comm. Math. Phys. 254 (2005), no. 2, 425-478. MR 2117633 (2006e:81263),
  • [BC11] Andrea Brini and Renzo Cavalieri, Open orbifold Gromov-Witten invariants of $ [\mathbb{C}^3/\mathbb{Z}_n]$: localization and mirror symmetry, Selecta Math. (N.S.) 17 (2011), no. 4, 879-933. MR 2861610 (2012i:14068),
  • [BCY10] Jim Bryan, Charles Cadman, and Ben Young, The orbifold topological vertex, Adv. Math. 229 (2012), no. 1, 531-595. MR 2854183,
  • [BG09] Jim Bryan and Tom Graber, The crepant resolution conjecture, Algebraic geometry--Seattle 2005. Part 1, Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., Providence, RI, 2009, pp. 23-42. MR 2483931 (2009m:14083)
  • [BGP08] Jim Bryan, Tom Graber, and Rahul Pandharipande, The orbifold quantum cohomology of $ \mathbb{C}^2/Z_3$ and Hurwitz-Hodge integrals, J. Algebraic Geom. 17 (2008), no. 1, 1-28. MR 2357679 (2008h:14056),
  • [CC09] Charles Cadman and Renzo Cavalieri, Gerby localization, $ Z_3$-Hodge integrals and the GW theory of $ [\mathbb{C}^3/Z_3]$, Amer. J. Math. 131 (2009), no. 4, 1009-1046. MR 2543921 (2010e:14051),
  • [CR04] Weimin Chen and Yongbin Ruan, A new cohomology theory of orbifold, Comm. Math. Phys. 248 (2004), no. 1, 1-31. MR 2104605 (2005j:57036),
  • [CR07] Tom Coates and Yongbin Ruan.
    Quantum cohomology and crepant resolutions: A conjecture, 2007.
  • [CR11] R. Cavalieri and D. Ross.
    Open Gromov-Witten theory and the crepant resolution conjecture.
    Preprint: math/1102.0717v1, 2011.
  • [DF05] Duiliu-Emanuel Diaconescu and Bogdan Florea, Localization and gluing of topological amplitudes, Comm. Math. Phys. 257 (2005), no. 1, 119-149. MR 2163571 (2006h:14064),
  • [FMN07] B. Fantechi, E. Mann, and F. Nironi.
    Smooth toric dm stacks.
    J. Reine Angew. Math., 2007.
  • [FP00] C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139 (2000), no. 1, 173-199. MR 1728879 (2000m:14057),
  • [GP99] T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487-518. MR 1666787 (2000h:14005),
  • [HKK$^+$03] Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipande, Richard Thomas, Cumrun Vafa, Ravi Vakil, and Eric Zaslow.
    Mirror Symmetry.
    AMS CMI, 2003. MR 2003030 (2004g:14042)
  • [Joh09] Paul D. Johnson, Equivariant Gromov-Witten theory of one dimensional stacks, ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)-University of Michigan. MR 2713030
  • [JPT08] P. Johnson, R. Pandharipande, and H.-H. Tseng, Abelian Hurwitz-Hodge integrals, Michigan Math. J. 60 (2011), no. 1, 171-198. MR 2785870 (2012c:14025),
  • [KL02] Sheldon Katz and Chiu-Chu Melissa Liu, Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc, Adv. Theor. Math. Phys. 5 (2001), no. 1, 1-49. MR 1894336 (2003e:14047)
  • [LLLZ09] Jun Li, Chiu-Chu Melissa Liu, Kefeng Liu, and Jian Zhou, A mathematical theory of the topological vertex, Geom. Topol. 13 (2009), no. 1, 527-621. MR 2469524 (2009i:14082),
  • [Mac95] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144 (96h:05207)
  • [MNOP06a] D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory. I, Compos. Math. 142 (2006), no. 5, 1263-1285. MR 2264664 (2007i:14061),
  • [MNOP06b] D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory. II, Compos. Math. 142 (2006), no. 5, 1286-1304. MR 2264665 (2007i:14062),
  • [MOOP11] D. Maulik, A. Oblomkov, A. Okounkov, and R. Pandharipande, Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds, Invent. Math. 186 (2011), no. 2, 435-479. MR 2845622 (2012h:14140),
  • [ORV06] Andrei Okounkov, Nikolai Reshetikhin, and Cumrun Vafa, Quantum Calabi-Yau and classical crystals, The unity of mathematics, Progr. Math., vol. 244, Birkhäuser Boston, Boston, MA, 2006, pp. 597-618. MR 2181817 (2006k:81342),
  • [Zon11] Z. Zong.
    Generalized Mariño-Vafa formula and local Gromov-Witten theory of orbi-curves.
    Preprint: math/1109.4992v1, 2011.

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Additional Information

Dustin Ross
Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523-1874
Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043

Received by editor(s): November 3, 2011
Received by editor(s) in revised form: March 20, 2012
Published electronically: November 4, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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