Mariño-Vafa formula and Hodge integral identities
Authors: Chiu-Chu Melissa Liu, Kefeng Liu and Jian Zhou
Published electronically: September 7, 2005
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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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
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
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