Journal of Algebraic Geometry

Author(s): Chiu-Chu Melissa Liu; Kefeng Liu; Jian Zhou
Journal: J. Algebraic Geom. 15 (2006), 379-398.
Posted: September 7, 2005
Retrieve article in: PDF DVI PostScript

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.

1.: 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)
2.: C. Faber, R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139 (2000), no. 1, 173-199. MR 1728879 (2000m:14057)
3.: C. Faber, R. Pandharipande, Hodge integrals, partition matrices, and the $\lambda_g$ conjecture, Ann. of Math. (2) 157 (2003), no. 1, 97-124. MR 1954265 (2004b:14095)
4.: E. Getzler, R. Pandharipande, Virasoro constraints and the Chern classes of the Hodge bundle, Nuclear Phys. B 530 (1998), no. 3, 701-714. MR 1653492 (2000b:14073)
5.: I.P. Goulden, D.M. Jackson, Combinatorial enumeration, John Wiley & Sons, 1983. MR 0702512 (84m:05002)
6.: 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)
7.: A. Givental, Gromov-Witten invariants and quantization of quadratic Hamiltonians, dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary, Mosc. Math. J. 1 (2001), no. 4, 551-568, 645. MR 1901075 (2003j:53138)
8.: T. Graber, R. Vakil, Hodge integrals and Hurwitz numbers via virtual localization, Compositio Math. 135 (2003), no. 1, 25-36. MR 1955162 (2004c:14108)
9.: E.-N. Ionel, T.H. Parker, Relative Gromov-Witten invariants, Ann. of Math. (2) 157 (2003), no. 1, 45-96. MR 1954264 (2004a:53112)
10.: E.-N. Ionel, T.H. Parker, The Symplectic Sum Formula for Gromov-Witten Invariants, Ann. of Math. (2) 159 (2004), no. 3, 935-1025. MR 2113018
11.: 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)
12.: 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)
13.: J. Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001), no. 3, 509-578. MR 1882667 (2003d:14066)
14.: J. Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199-293. MR 1938113 (2004k:14096)
15.: 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. MR 2067471
16.: 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
17.: I.G. MacDonald, Symmetric functions and Hall polynomials, 2nd edition. Clarendon Press, 1995. MR 1354144 (96h:05207)
18.: M. Mariño, C. Vafa, Framed knots at large , Orbifolds in mathematics and physics (Madison, WI, 2001), 185-204, Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002. MR 1950947 (2004c:57024)
19.: D. Mumford, Towards an enumerative geometry of the moduli space of curves, in Arithmetic and geometry, vol. II, Progr. Math., 36, Birkhauser Boston, Boston, 1983, pp. 271-328. MR 0717614 (85j:14046)
20.: A. Okounkov, R. Pandharipande, Virasoro constraints for target curves, math. AG/0308097.
21.: A. Okounkov, R. Pandharipande, Hodge integrals and invariants of the unknots, Geom. Topol. 8 (2004), 675-699. MR 2057777
22.: J. Zhou, Hodge integrals, Hurwitz numbers, and symmetric groups, math.AG/0308024.

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

PII: S 1056-3911(05)00419-4
Received by editor(s): March 15, 2005
Posted: September 7, 2005

Previous issue \| Table of contents \| Next issue Recently posted articles \| Most recent issue \| All issues