Mariño-Vafa formula and Hodge integral identities

Liu, Chiu-Chu Melissa; Liu, Kefeng; Zhou, Jian

doi:10.1090/S1056-3911-05-00419-4

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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Abstract | 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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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