On the recursion formula for double Hurwitz numbers
Author:
Shengmao Zhu
Journal:
Proc. Amer. Math. Soc. 140 (2012), 37493760
MSC (2010):
Primary 14H10; Secondary 05E05
Published electronically:
March 12, 2012
MathSciNet review:
2944715
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Abstract: In this paper, we will give a recursion formula for double Hurwitz numbers by the cutjoin analysis. This recursion formula can be considered as a generalized version of the recursion formula for simple Hurwitz numbers derived by Mulase and Zhang. As a direct application, we get a polynomial identity for GouldenJacksonVakil's conjectural intersection numbers and an explicit recursion formula for the computation of these intersection numbers with only classes.
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 2.
 T. Ekedahl, S. Lando, M. Shapiro, and A. Vainshtein, Hurwitz numbers and intersections on moduli spaces of curves, Invent. Math. 146 (2001), 297327. MR 1864018 (2002j:14034)
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 4.
 C. Faber and R. Pandharipande, Hodge integrals, partition matrices, and the conjecture, Ann. of Math. (2) 157 (2003), 97124. MR 1954265 (2004b:14095)
 5.
 C. Faber and R. Pandharipande, Logarithmic series and Hodge integrals in the tautological ring (with an appendix by D. Zagier), Michigan Math. J. 48 (2000), 215252. MR 1786488 (2002e:14041)
 6.
 I.P. Goulden and D.M. Jackson,Transitive factorisations into transpositions and holomorphic mappings on the sphere, Proc. Amer. Math. Soc. 125 (1997), 5160. MR 1396978 (97j:05007)
 7.
 I.P. Goulden, D.M. Jackson and R. Vakil, Towards the geometry of double Hurwitz numbers, Adv. Math. 198 (2005), 4392. MR 2183250 (2006i:14023)
 8.
 I.P. Goulden, D.M. Jackson and R. Vakil, A short proof of the conjecture without GromovWitten theory: Hurwitz theory and the moduli of curves, J. Reine Angew. Math. 637 (2009), 175191. MR 2599085 (2011f:14087)
 9.
 E. Getzler and R. Pandharipande, Virasoro constraints and the Chern classes of the Hodge bundle, Nuclear Phys. B 530 (1998), 701714. MR 1653492 (2000b:14073)
 10.
 M. Kazarian and S. Lando, An algebrogeometric proof of Witten's conjecture, J. Amer. Math. Soc. 20 (2007), 10791089. MR 2328716 (2008d:14055)
 11.
 Y.S. Kim and K. Liu, A simple proof of Witten conjecture through localization, preprint, arXiv:math/0508384 [math.AG] (2005).
 12.
 A.M. Li, G. Zhao and Q. Zheng, The number of ramified coverings of a Riemann surface by Riemann surface, Commun. Math. Phys. 213 (2000), 685696 . MR 1785434 (2001i:14078)
 13.
 K. Liu and H. Xu, New results of intersection numbers on moduli spaces of curves, Proc. Natl. Acad. Sci. USA 104 (2007), 1389613900. MR 2348851 (2008f:14041)
 14.
 K. Liu and H. Xu, A proof of the Faber intersection number conjecture, J. Differential Geom. 83 (2009), 313335. MR 2577471 (2011d:14051)
 15.
 M. Mulase and Naizhen Zhang, Polynomial recursion for linear Hodge integrals, Commun. Number Theory Phys. 4 (2010), 267294. MR 2725053
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 A. Okounkov and R. Pandharipande, GromovWitten theory, Hurwitz numbers, and matrix models, I, Proc. Symposia Pure Math. 80 (2009), 325414. MR 2483941 (2009k:14111)
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 E. Witten, Twodimensional gravity and intersection theory on moduli space, Surv. Differ. Geom. 1 (1991), 243310. MR 1144529 (93e:32028)
 18.
 H. Xu, Hodge integrals on moduli spaces of curves, thesis submitted for the degree of Doctor of Philosophy of Zhejiang University, 2009.
 19.
 J. Zhou, Hodge integrals, Hurwitz numbers, and symmetric groups, arXiv:math/0308024.
 20.
 J. Zhou, On recursion relation for Hodge integrals from the cutandjoin equations, preprint, 2009.
 21.
 S. Zhu, Hodge integral with one class, Sci. China Math. 55 (2012), doi:10.1007/S1142501143137
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Additional Information
Shengmao Zhu
Affiliation:
Department of Mathematics and Center of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang 310027, People’s Republic of China
Email:
zhushengmao@gmail.com
DOI:
http://dx.doi.org/10.1090/S000299392012112353
Keywords:
Hurwitz numbers,
moduli space,
cutjoin,
recursion
Received by editor(s):
November 30, 2010
Received by editor(s) in revised form:
April 28, 2011
Published electronically:
March 12, 2012
Communicated by:
Lev Borisov
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
