Mariño-Vafa formula and Hodge integral identities
Authors:
Chiu-Chu Melissa Liu, Kefeng Liu and Jian Zhou
Journal:
J. Algebraic Geom. 15 (2006), 379-398
DOI:
https://doi.org/10.1090/S1056-3911-05-00419-4
Published electronically:
September 7, 2005
MathSciNet review:
2199062
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References |
Additional Information
Abstract: We derive some Hodge integral identities by taking various limits of the Mariño-Vafa formula using the cut-and-join equation. These identities include the formula of general $\lambda _g$-integrals, the formula of $\lambda _{g-1}$-integrals on ${\overline {\mathcal {M}}}_{g,1}$, the formula of cubic $\lambda$ integrals on ${\overline {\mathcal {M}}}_g$, and the ELSV formula relating Hurwitz numbers and Hodge integrals. In particular, our proof of the MV formula by the cut-and-join equation leads to a new and simple proof of the $\lambda _g$ conjecture. We also present a proof of the ELSV formula completely parallel to our proof of the Mariño-Vafa formula.
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Fab-Pan1 C. Faber, R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139 (2000), no. 1, 173–199.
Fab-Pan2 C. Faber, R. Pandharipande, Hodge integrals, partition matrices, and the $\lambda _g$ conjecture, Ann. of Math. (2) 157 (2003), no. 1, 97-124.
Get-P E. Getzler, R. Pandharipande, Virasoro constraints and the Chern classes of the Hodge bundle, Nuclear Phys. B 530 (1998), no. 3, 701–714.
Gou-Jac I.P. Goulden, D.M. Jackson, Combinatorial enumeration, John Wiley & Sons, 1983.
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Gra-Vak T. Graber, R. Vakil, Hodge integrals and Hurwitz numbers via virtual localization, Compositio Math. 135 (2003), no. 1, 25–36.
Ion-Par1 E.-N. Ionel, T.H. Parker, Relative Gromov-Witten invariants, Ann. of Math. (2) 157 (2003), no. 1, 45–96.
Ion-Par E.-N. Ionel, T.H. Parker, The Symplectic Sum Formula for Gromov-Witten Invariants, Ann. of Math. (2) 159 (2004), no. 3, 935–1025.
Li-Rua A.-M. Li, Y. Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds, Invent. Math. 145 (2001), no. 1, 151–218.
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Li2 J. Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199–293.
LLZ1 C.-C. Liu, K. Liu, J. Zhou, On a proof of a conjecture of Mariño-Vafa on Hodge Integrals, Math. Res. Lett. 11 (2004), no. 2-3, 259–272.
LLZ 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.
Mac I.G. MacDonald, Symmetric functions and Hall polynomials, 2nd edition. Clarendon Press, 1995.
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Zho J. Zhou, Hodge integrals, Hurwitz numbers, and symmetric groups, arXiv:math.AG/0308024.
Additional Information
Chiu-Chu Melissa Liu
Affiliation:
Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138 and Center of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang 310027, China
Address at time of publication:
Department of Mathematics, Northwestern University, Evanston, Illinois 60208-2370
MR Author ID:
691648
Email:
ccliu@math.harvard.edu, ccliu@math.northwestern.edu
Kefeng Liu
Affiliation:
Center of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang 310027, China and Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095-1555
MR Author ID:
327618
Email:
liu@cms.zju.edu.cn; liu@math.ucla.edu
Jian Zhou
Affiliation:
Center of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang 310027, China and Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
Email:
jzhou@math.tsinghua.edu.cn
Received by editor(s):
March 15, 2005
Published electronically:
September 7, 2005