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A formula of two-partition Hodge integrals


Authors: Chiu-Chu Melissa Liu, Kefeng Liu and Jian Zhou
Journal: J. Amer. Math. Soc. 20 (2007), 149-184
MSC (2000): Primary 14N35, 53D45; Secondary 57M27
DOI: https://doi.org/10.1090/S0894-0347-06-00541-8
Published electronically: August 1, 2006
MathSciNet review: 2257399
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Abstract: Motivated by the Mariño-Vafa formula of Hodge integrals and physicists' predictions on local Gromov-Witten invariants of toric Fano surfaces in a Calabi-Yau threefold, the third author conjectured a formula of certain Hodge integrals in terms of certain Chern-Simons invariants of the Hopf link. We prove this formula by virtual localization on moduli spaces of relative stable morphisms.


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  • 1. M. Aganagic, A. Klemm, M. Marino, C. Vafa, The topological vertex, Comm. Math. Phys. 254 (2005), no. 2, 425-478. MR 2117633 (2006e:81263)
  • 2. M. Aganagic, M. Marino, C. Vafa, All loop topological string amplitudes from Chern-Simons theory, Comm. Math. Phys. 247 (2004), no. 2, 467-512. MR 2063269 (2005g:81284)
  • 3. J. Bryan, R. Pandharipande, Curves in Calabi-Yau $ 3$-folds and topological quantum field theory, Duke Math. J. 126 (2005), no. 2, 369-396. MR 2115262 (2005k:14117)
  • 4. D.-E. Diaconescu, B. Florea, Localization and gluing of topological amplitudes, Comm. Math. Phys. 257 (2005), no. 1, 119-149. MR 2163571
  • 5. R. Dijkgraaf, Mirror symmetry and elliptic curves, The moduli space of curves, R. Dijkgraaf, G. van der Geer (editors), Prog. in Math., 129, Birkhäuser, 1995. MR 1363055 (96m:14072)
  • 6. T. Ekedahl, S. Lando, M. Shapiro, A. Vainshtein, Hurwitz numbers and intersections on moduli spaces of curves. Invent. Math. 146 (2001), no. 2, 297-327. MR 1864018 (2002j:14034)
  • 7. B. Fantechi, R. Pandharipande, Stable maps and branch divisors, Compositio Math. 130 (2002), no. 3, 345-364. MR 1887119 (2003a:14039)
  • 8. I.P. Goulden, D.M. Jackson, A. Vainshtein, The number of ramified coverings of the sphere by the torus and surfaces of higher genera, Ann. of Comb. 4 (2000), 27-46. MR 1763948 (2002b:14067)
  • 9. T. Graber, R. Pandharipande, Localization of virtual classes. Invent. Math. 135 (1999), no. 2, 487-518. MR 1666787 (2000h:14005)
  • 10. T. Graber, R. Vakil, Hodge integrals and Hurwitz numbers via virtual localization. Compositio Math. 135 (2003), no. 1, 25-36. MR 1955162 (2004c:14108)
  • 11. T. Graber, R. Vakil, Relative virtual localization and vanishing of tautological classes on moduli spaces of curves, Duke Math. J. 130 (2005), no. 1, 1-37. MR 2176546
  • 12. A. Iqbal, All genus topological amplitudes and $ 5$-brane webs as Feynman diagrams, preprint, hep-th/0207114.
  • 13. E.-N. Ionel, T. Parker, Relative Gromov-Witten invariants, Ann. of Math. (2) 157 (2003), no. 1, 45-96. MR 1954264 (2004a:53112)
  • 14. E.-N. Ionel, T. Parker, The symplectic sum formula for Gromov-Witten invariants, Ann. of Math. (2) 159 (2004), no. 3, 935-1025. MR 2113018 (2006b:53110)
  • 15. S. Katz, C.-C. Liu, Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc, Adv. Theor. Math. Phys. 5 (2001), 1-49. MR 1894336 (2003e:14047)
  • 16. M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function. Comm. Math. Phys. 147 (1992), no. 1, 1-23. MR 1171758 (93e:32027)
  • 17. A.M. Li, Y. Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau $ 3$-folds, Invent. Math. 145 (2001), no. 1, 151-218. MR 1839289 (2002g:53158)
  • 18. A.M. Li, G. Zhao, Q. Zheng, The number of ramified coverings of a Riemann surface by Riemann surface, Comm. Math. Phys. 213 (2000), no. 3, 685-696. MR 1785434 (2001i:14078)
  • 19. J. Li, Stable morphisms to singular schemes and relative stable morphisms, J. Diff. Geom. 57 (2001), 509-578. MR 1882667 (2003d:14066)
  • 20. J. Li, A degeneration formula of Gromov-Witten invariants, J. Diff. Geom. 60 (2002), 199-293. MR 1938113 (2004k:14096)
  • 21. J. Li, Lecture notes on relative GW-invariants, Intersection theory and moduli, 41-96 (electronic), ICTP Lect. Notes, XIX, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004. MR 2172495
  • 22. J. Li, C.-C. Liu, K. Liu, J. Zhou, A mathematical theory of the topological vertex, preprint, math.AG/0408426.
  • 23. C.-C. Liu, K. Liu, J. Zhou, A proof of a conjecture of Mariño-Vafa on Hodge Integrals, J. Differential Geom. 65 (2003), no. 2, 289-340. MR 2058264 (2005m:14104)
  • 24. I.G. MacDonald, Symmetric functions and Hall polynomials, Second edition. With contributions by A. Zelevinsky. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. MR 1354144 (96h:05207)
  • 25. M. Mariño, C. Vafa, Framed knots at large $ N$, Orbifolds in mathematics and physics (Madison, WI, 2001), 185-204, Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002. MR 1950947 (2004c:57024)
  • 26. H.R. Morton, S.G. Lukac, The HOMFLY polynomial of the decorated Hopf link, J. Knot Theory Ramifications 12 (2003), no. 3, 395-416. MR 1983094 (2004d:57012)
  • 27. A. Okounkov, Toda equations for Hurwitz numbers, Math. Res. Lett. 7 (2000), no. 4, 447-453. MR 1783622 (2001i:14047)
  • 28. A. Okounkov, R. Pandharipande, Gromov-Witten theory, Hurwitz theory, and complete cycles, Ann. of Math. (2) 163 (2006), no.2, 517-560. MR 2199225
  • 29. A. Okounkov, R. Pandharipande, Hodge integrals and invariants of the unknots, Geom. Topol. 8 (2004), 675-699 (electronic). MR 2057777 (2006c:14041)
  • 30. E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351-399. MR 0990772 (90h:57009)
  • 31. E. Witten, Two-dimensional gravity and intersection theory on moduli space. Surveys in differential geometry (Cambridge, MA, 1990), 243-310, Lehigh Univ., Bethlehem, PA, 1991. MR 1144529 (93e:32028)
  • 32. J. Zhou, Hodge integrals, Hurwitz numbers, and symmetric groups, preprint, math.AG/0308024, to appear in Comm. Anal. Geom.
  • 33. J. Zhou, Mariño-Vafa formula and BPS numbers in local $ \mathbb{P}^2$ and $ \mathbb{P}^1 \times \mathbb{P}^1$ geometry, preprint, submitted.
  • 34. J. Zhou, A conjecture on Hodge integrals, preprint, math.AG/0310282.
  • 35. J. Zhou, Localizations on moduli spaces and free field realizations of Feynman rules, preprint, math.AG/0310283.

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Additional Information

Chiu-Chu Melissa Liu
Affiliation: Center of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang 310027, People’s Republic of China and Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Address at time of publication: Department of Mathematics, Columbia University, New York, New York, 10027
Email: ccliu@math.columbia.edu

Kefeng Liu
Affiliation: Center of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang 310027, People’s Republic of China and Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095-1555
Email: liu@cms.zju.edu.cn, liu@math.ucla.edu

Jian Zhou
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China and Center of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang 310027, People’s Republic of China
Email: jzhou@math.tsinghua.edu.cn

DOI: https://doi.org/10.1090/S0894-0347-06-00541-8
Keywords: Hodge integrals, localization, Gromov-Witten invariants, Hopf link
Received by editor(s): January 19, 2005
Published electronically: August 1, 2006
Article copyright: © Copyright 2006 American Mathematical Society

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