Localization and gluing of orbifold amplitudes: The Gromov-Witten orbifold vertex
HTML articles powered by AMS MathViewer
- by Dustin Ross PDF
- Trans. Amer. Math. Soc. 366 (2014), 1587-1620 Request permission
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.References
- Mina Aganagic, Albrecht Klemm, Marcos Mariño, and Cumrun Vafa, The topological vertex, Comm. Math. Phys. 254 (2005), no. 2, 425–478. MR 2117633, DOI 10.1007/s00220-004-1162-z
- Andrea Brini and Renzo Cavalieri, Open orbifold Gromov-Witten invariants of $[\Bbb C^3/\Bbb Z_n]$: localization and mirror symmetry, Selecta Math. (N.S.) 17 (2011), no. 4, 879–933. MR 2861610, DOI 10.1007/s00029-011-0060-4
- Jim Bryan, Charles Cadman, and Ben Young, The orbifold topological vertex, Adv. Math. 229 (2012), no. 1, 531–595. MR 2854183, DOI 10.1016/j.aim.2011.09.008
- 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, DOI 10.1090/pspum/080.1/2483931
- Jim Bryan, Tom Graber, and Rahul Pandharipande, The orbifold quantum cohomology of $\Bbb C^2/Z_3$ and Hurwitz-Hodge integrals, J. Algebraic Geom. 17 (2008), no. 1, 1–28. MR 2357679, DOI 10.1090/S1056-3911-07-00467-5
- Charles Cadman and Renzo Cavalieri, Gerby localization, $Z_3$-Hodge integrals and the GW theory of $[\Bbb C^3/Z_3]$, Amer. J. Math. 131 (2009), no. 4, 1009–1046. MR 2543921, DOI 10.1353/ajm.0.0063
- Weimin Chen and Yongbin Ruan, A new cohomology theory of orbifold, Comm. Math. Phys. 248 (2004), no. 1, 1–31. MR 2104605, DOI 10.1007/s00220-004-1089-4
- Tom Coates and Yongbin Ruan. Quantum cohomology and crepant resolutions: A conjecture, 2007.
- R. Cavalieri and D. Ross. Open Gromov-Witten theory and the crepant resolution conjecture. Preprint: math/1102.0717v1, 2011.
- Duiliu-Emanuel Diaconescu and Bogdan Florea, Localization and gluing of topological amplitudes, Comm. Math. Phys. 257 (2005), no. 1, 119–149. MR 2163571, DOI 10.1007/s00220-005-1323-8
- B. Fantechi, E. Mann, and F. Nironi. Smooth toric dm stacks. J. Reine Angew. Math., 2007.
- C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139 (2000), no. 1, 173–199. MR 1728879, DOI 10.1007/s002229900028
- T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518. MR 1666787, DOI 10.1007/s002220050293
- Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipande, Richard Thomas, Cumrun Vafa, Ravi Vakil, and Eric Zaslow, Mirror symmetry, Clay Mathematics Monographs, vol. 1, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2003. With a preface by Vafa. MR 2003030
- 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
- P. Johnson, R. Pandharipande, and H.-H. Tseng, Abelian Hurwitz-Hodge integrals, Michigan Math. J. 60 (2011), no. 1, 171–198. MR 2785870, DOI 10.1307/mmj/1301586310
- 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, DOI 10.4310/ATMP.2001.v5.n1.a1
- 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, DOI 10.2140/gt.2009.13.527
- 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
- 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, DOI 10.1112/S0010437X06002302
- 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, DOI 10.1112/S0010437X06002314
- 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, DOI 10.1007/s00222-011-0322-y
- 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, DOI 10.1007/0-8176-4467-9_{1}6
- Z. Zong. Generalized Mariño-Vafa formula and local Gromov-Witten theory of orbi-curves. Preprint: math/1109.4992v1, 2011.
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
- Email: ross@math.colostate.edu, dustyr@umich.edu
- Received by editor(s): November 3, 2011
- Received by editor(s) in revised form: March 20, 2012
- Published electronically: November 4, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 1587-1620
- MSC (2010): Primary 14N35; Secondary 05A15
- DOI: https://doi.org/10.1090/S0002-9947-2013-05835-7
- MathSciNet review: 3145743