Hurwitz numbers and intersections on moduli spaces of curves

Ekedahl, Torsten; Lando, S. K.; Shapiro, Michael; Vainshtein, Alek

doi:10.1007/s002220100164

Cited by 287 publications

(365 citation statements)

References 28 publications

Supporting

Mentioning

350

Contrasting

Unclassified

Order By: Relevance

“…Let Ξ be the hyperplane class of P r associated to the canonical hyperplane bundle. Then the simple Hurwitz numbers can be represented as Gromov-Witten integrals [52]- [54] …”

Section: Gross-taylor Expansion As a Topological String Theorymentioning

confidence: 99%

See 1 more Smart Citation

Topological strings and largeNphase transitions II: chiral expansion ofq-deformed Yang-Mills theory

Caporaso

Cirafici

Pasquetti

et al. 2006

J. High Energy Phys.

View full text Add to dashboard Cite

We continue our study of the large N phase transition in q-deformed YangMills theory on the sphere and its role in connecting topological strings to black hole entropy. We study in detail the chiral theory defined in terms of uncoupled single U (N ) representations at large N and write down the resulting partition function by means of the topological vertex. The emergent toric geometry has three Kähler parameters, one of which corresponds to the expected fibration over P 1 . By taking a suitable double-scaling limit we recover the chiral Gross-Taylor string expansion. To analyse the phase transition we construct a matrix model which describes the chiral gauge theory. It has three distinct phases, one of which should be described by the closed topological string expansion. We verify this expectation by explicit comparison between the matrix model and the chiral topological string free energies. We also show that the critical point in the pertinent phase of the matrix model corresponds to a divergence of the topological string perturbation series.

show abstract

“…Let Ξ be the hyperplane class of P r associated to the canonical hyperplane bundle. Then the simple Hurwitz numbers can be represented as Gromov-Witten integrals [52]- [54] …”

Section: Gross-taylor Expansion As a Topological String Theorymentioning

confidence: 99%

“…Let E g → M g,n be the rank g Hodge bundle, and denote the corresponding Chern classes by λ k = c k (E g ) ∈ H 2k ( M g,n , Q). With λ 0 := 1, the localization formula then reads [52]- [54] …”

Section: Gross-taylor Expansion As a Topological String Theorymentioning

confidence: 99%

Topological strings and largeNphase transitions II: chiral expansion ofq-deformed Yang-Mills theory

Caporaso

Cirafici

Pasquetti

et al. 2006

J. High Energy Phys.

View full text Add to dashboard Cite

show abstract

“…• the ELSV formula [4,5] relating Hurwitz numbers to the intersection theory on moduli spaces, • the relationship between Hurwitz numbers and integrable hierarchies conjectured by Pandharipande [14] and proved, in a stronger form, by Okounkov [12].…”

Section: Introductionmentioning

confidence: 99%

“…The ELSV formula [4,5] expresses the Hurwitz numbers in terms of Hodge integrals over the moduli spaces of stable complex curves:…”

Section: Introductionmentioning

confidence: 99%

An algebro-geometric proof of Witten’s conjecture

Kazarian

Lando

2007

J. Amer. Math. Soc.

126

View full text Add to dashboard Cite

show abstract

“…The Join-Cut equation also extends in a straightforward way to the case of arbitrary genus. In arbitrary genus, the corresponding numbers (in which one fixed branch point has arbitrary ramification, and all others are simple) are given by the ELSV formula [2], as a Hodge integral. However, we know of no simple combinatorial or geometric proof of Theorem 6.…”

Section: Outlinementioning

confidence: 99%

A Simple Recurrence for Covers of the Sphere With Branch Points of Arbitrary Ramification

Goulden

Serrano

2006

Ann. Comb.

View full text Add to dashboard Cite

The problem of counting ramified covers of a Riemann surface up to homeomorphism was proposed by Hurwitz in the late 1800's. This problem translates combinatorially into factoring a permutation of specified cycle type, with certain conditions on the cycle types of the factors, such as minimality and transitivity.Goulden and Jackson have given a proof for the number of minimal, transitive factorizations of a permutation into transpositions. This proof involves a partial differential equation for the generating series, called the Join-Cut equation. Recently, Bousquet-Mélou and Schaeffer have found the number of minimal, transitive factorizations of a permutation into arbitrary unspecified factors. This was proved by a purely combinatorial argument, based on a direct bijection between factorizations and certain objects called m-Eulerian trees.In this paper, we give a simple partial differential equation for Bousquet-Mélou and Schaeffer's generating series, and for Goulden and Jackson's generating series, as well as a new proof of the result by Bousquet-Mélou and Schaeffer. We apply algebraic methods based on Lagrange's theorem, and combinatorial methods based on a new use of Bousquet-Mélou and Schaeffer's m-Eulerian trees.

show abstract

Hurwitz numbers and intersections on moduli spaces of curves

Cited by 287 publications

References 28 publications

Topological strings and largeNphase transitions II: chiral expansion ofq-deformed Yang-Mills theory

Topological strings and largeNphase transitions II: chiral expansion ofq-deformed Yang-Mills theory

An algebro-geometric proof of Witten’s conjecture

A Simple Recurrence for Covers of the Sphere With Branch Points of Arbitrary Ramification

Contact Info

Product

Resources

About