Hodge integrals, Hurwitz numbers, and Symmetric Groups
@article{Zhou2003HodgeIH, title={Hodge integrals, Hurwitz numbers, and Symmetric Groups}, author={Jian Zhou}, journal={arXiv: Algebraic Geometry}, year={2003} }
We prove some combinatorial results related to a formula on Hodge integrals conjectured by Mari\~no and Vafa. These results play important roles in the proof and applications of this formula by the author jointly with Chiu-Chu Melissa Liu and Kefeng Liu. We also compare with some related results on Hurwitz numbers and obtain some closed expressions for the generating series of Hurwitz numbers and the related Hodge integrals.
38 Citations
A Conjecture on Hodge Integrals
- Mathematics
- 2003
We propose a conjectural formula expressing the generating series of some Hodge integrals in terms of representation theory of Kac-Moody algebras. Such generating series appear in calculations of…
A Proof of a Conjecture of Mariño-Vafa on Hodge Integrals
- Mathematics
- 2003
We prove a remarkable formula for Hodge integrals conjectured by Marino and Vafa, 2002, based on large N duality, using functorial virtual localization on certain moduli spaces of relative stable…
A Proof of a Conjecture of Marino-Vafa on Hodge Integrals
- Mathematics
- 2003
We prove a remarkable formula for Hodge integrals conjectured by Marino and Vafa based on large N duality, using functorial virtual localization on certain moduli spaces of relative stable morphisms.
Hodge Integrals and Integrable Hierarchies
- Mathematics
- 2003
We show that the generating series of some Hodge integrals involving one or two partitions are τ-functions of the KP hierarchy or the 2-Toda hierarchy, respectively. We also reformulate the results…
Some Results of the Mari˜no -vafa Formula
- Mathematics
- 2006
In this paper we derive some new Hodge integral identities by taking limits of the Mariño-Vafa formula. These identities include the formula of λ 1 λ g-integral on M g,1 , the vanishing result of λ g…
KP hierarchy for Hurwitz-type cohomological field theories
- Mathematics
- 2021
Abstract. We generalise a result of Kazarian regarding Kadomtsev-Petviashvili integrability for single Hodge integrals to general cohomological field theories related to Hurwitz-type counting…
Curve counting and instanton counting
- Mathematics
- 2003
We prove some combinatorial results required for the proof of the following conjecture of Nekrasov: The generating function of closed string invariants in local Calabi-Yau geometries obtained by…
A formula of two-partition Hodge integrals
- Mathematics
- 2003
JMg,n where tpi = ci(L?) is the first Chern class of L?, and Xj = Cj(E) is the j-th Chern class of the Hodge bundle. The study of Hodge integrals is an important part of intersection theory on Mg,n>…
Polynomial recursion formula for linear Hodge integrals
- Mathematics
- 2009
We establish a polynomial recursion formula for linear Hodge integrals. It is obtained as the Laplace transform of the cut-and-join equation for the simple Hurwitz numbers. We show that the recursion…
Computing Hodge Integrals with One Λ-class
- Mathematics
- 2008
where ψi is the first Chern class of the cotangent line bundle at the i-th marked point, and λ1, · · · , λg are the Chern classes of the Hodge bundle. Hodge integrals arise naturally in the…
References
SHOWING 1-10 OF 31 REFERENCES
Hodge Integrals and Hurwitz Numbers via Virtual Localization
- MathematicsCompositio Mathematica
- 2003
We give another proof of Ekedahl, Lando, Shapiro, and Vainshtein's remarkable formula expressing Hurwitz numbers (counting covers of P1 with specified simple branch points, and specified branching…
Mariño-Vafa formula and Hodge integral identities
- Mathematics
- 2003
Based on string duality Marino and Vafa [10] conjectured a closed formula on certain Hodge integrals in terms of representations of symmetric groups. This formula was first explicitly written down by…
Hurwitz numbers and intersections on moduli spaces of curves
- Mathematics
- 2000
This article is an extended version of preprint math.AG/9902104. We find an explicit formula for the number of topologically different ramified coverings of a sphere by a genus g surface with only…
A Proof of a Conjecture of Mariño-Vafa on Hodge Integrals
- Mathematics
- 2003
We prove a remarkable formula for Hodge integrals conjectured by Marino and Vafa, 2002, based on large N duality, using functorial virtual localization on certain moduli spaces of relative stable…
Quadratic Recursion Relations of Hodge Integrals via Localization
- Mathematics
- 2003
We derive some quadratic rccursion relations for some Hodge integrals by virtual localization and obtain many closed formulas.We apply our formulas to the local geometry of toric Fano surfaces in a…
A Proof of a Conjecture of Marino-Vafa on Hodge Integrals
- Mathematics
- 2003
We prove a remarkable formula for Hodge integrals conjectured by Marino and Vafa based on large N duality, using functorial virtual localization on certain moduli spaces of relative stable morphisms.
Some closed formulas and conjectures for Hodge integrals
- Mathematics
- 2003
We announce the proof of some closed formulas for Hodge integrals, including some low degree cases of a remarkable formula conjectured by Marino and Vafa based on string duality, and a conjecture by…
Hodge integrals and Gromov-Witten theory
- Mathematics
- 1998
Integrals of the Chern classes of the Hodge bundle in Gromov-Witten theory are studied. We find a universal system of differential equations which determines the generating function of these…
New Trends in Algebraic Geometry: Algorithms for computing intersection numbers on moduli spaces of curves, with an application to the class of the locus of Jacobians
- Mathematics
- 1997
We describe algorithms for computing the intersection numbers of divisors and of Chern classes of the Hodge bundle on the moduli spaces of stable pointed curves. We also discuss the implementations…