Think globally, compute locally

Bouchard, Vincent; Eynard, Bertrand

doi:10.1007/jhep02(2013)143

Cited by 89 publications

(161 citation statements)

References 51 publications

Supporting

Mentioning

159

Contrasting

Unclassified

Order By: Relevance

“…Remark 10.1. It is natural to expect that, if branch points of higher order occur, the higher order version of the topological recursion relations introduced in [17] holds, and that the ideas of [17] together with our intermediate results may be used to prove the corresponding generalization. However, for the sake of brevity, we do not address this here, leaving it rather as an open problem for future work.…”

Section: Topological Recursionmentioning

confidence: 95%

Weighted Hurwitz Numbers and Topological Recursion

Alexandrov

Chapuy

Eynard

et al. 2020

Commun. Math. Phys.

View full text Add to dashboard Cite

The KP and 2D Toda τ -functions of hypergeometric type that serve as generating functions for weighted single and double Hurwitz numbers are related to the topological recursion programme. A graphical representation of such weighted Hurwitz numbers is given in terms of weighted constellations. The associated classical and quantum spectral curves are derived, and these are interpreted combinatorially in terms of the graphical model. The pair correlators are given a finite Christoffel-Darboux representation and determinantal expressions are obtained for the multipair correlators. The genus expansion of the multicurrent correlators is shown to provide generating series for weighted Hurwitz numbers of fixed ramification profile lengths. The WKB series for the Baker function is derived and used to deduce the loop equations and the topological recursion relations. * graphs such as maps, dessins d'enfants, or more generally, constellations [54]. There has also been important progress in relating Hurwitz numbers to other classes of enumerative geometric invariants [5,19,23,30,37,51,61,64,65,68] and matrix models [1,2,4,9,10,15,19,22,57,60,64,65].A key development was the identification by Pandharipande [68] and Okounkov [61] that certain special τ -functions for integrable hierarchies of the KP and 2D Toda type may serve as generating functions for simple (single and double) Hurwitz numbers (i.e., those for which all branch points, with the possible exception of one, or two, have simple ramification profiles). It was shown subsequently [9,10,38,39,41,44] that all weighted (single and double) Hurwitz numbers have KP or 2D Toda τ -functions of the special hypergeometric type [63, 66, 67] as generating functions.An alternative approach, particularly useful for studying genus dependence and recursive relations between the invariants involved [37], consists of using multicurrent correlators as generating functions for weighted Hurwitz numbers having a fixed ramification profile length n. These may be defined in a number of equivalent ways: either as the coefficients in multivariable Taylor series expansions of the τ -function about suitably defined n-parameter families of evaluation points, in terms of pair correlators, or as fermionic expectation values of products of current operators evaluated at n points [7].A very efficient way of computing Hurwitz numbers, which provides strong results about their structure, follows from the method of Topological Recursion (TR), introduced by Eynard and Orantin in [27]. This approach, originally inspired by results arising naturally in random matrix theory [24], has been shown applicable to many enumerative geometry problems, such as the counting of maps [25] or computation of Gromov-Witten invariants [18]. It has received a great deal of attention in recent years and found to have many far-reaching implications. The fact that simple Hurwitz numbers satisfy the TR relations was conjectured by Bouchard and Mariño [19], and proved in [15,26]. This provides an algorithm that allows them to...

show abstract

Section: Topological Recursionmentioning

confidence: 95%

Weighted Hurwitz Numbers and Topological Recursion

Alexandrov

Chapuy

Eynard

et al. 2020

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…Moreover, starting from these equations it should be possible to obtain a topological recursion like formula. Such a recursion formula certainly looks like the Bouchard-Eynard topological recursion formula introduced in [BHL + 14, BE13]. Establishing such a formula strongly depends on the analytic properties of the W g,n as well as the geometric information contained in W 0,1 and W 0,2 .…”

Section: Abmentioning

confidence: 98%

Schwinger–Dyson and loop equations for a product of square Ginibre random matrices

Dartois

Forrester

2020

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

In this paper, we study the product of two complex Ginibre matrices and the loop equations satisfied by their resolvents (i.e. the Stieltjes transform of the correlation functions). We obtain using Schwinger-Dyson equation (SDE) techniques the general loop equations satisfied by the resolvents. In order to deal with the product structure of the random matrix of interest, we consider SDEs involving the integral of higher derivatives. One of the advantage of this technique is that it bypasses the reformulation of the problem in terms of singular values. As a byproduct of this study we obtain the large N limit of the Stieltjes transform of the 2-point correlation function, as well as the first correction to the Stieltjes transform of the density, giving us access to corrections to the smoothed density. In order to pave the way for the establishment of a topological recursion formula we also study the geometry of the corresponding spectral curve. This paper also contains explicit results for different resolvents and their corrections.

show abstract

“…Observe that the mechanism behind the cut-and-join recursion is in some sense local. In the interpretation of Hurwitz numbers as enumerations of transitive factorisations, the recursion is not sensitive to the permutation τ 0 in equation (4). So using the parameter s allows us to express the cut-andjoin recursion identically in the case of simple Hurwitz numbers [25], orbifold Hurwitz numbers [12,5], as well as double Hurwitz numbers [35].…”

Section: Remarkmentioning

confidence: 99%

Towards the topological recursion for double Hurwitz numbers

Do¹,

Karev

2018

Proceedings of Symposia in Pure Mathematics

View full text Add to dashboard Cite

Single Hurwitz numbers enumerate branched covers of the Riemann sphere with specified genus, prescribed ramification over infinity, and simple branching elsewhere. They exhibit a remarkably rich structure. In particular, they arise as intersection numbers on moduli spaces of curves and are governed by the topological recursion of Chekhov, Eynard and Orantin. Double Hurwitz numbers are defined analogously, but with prescribed ramification over both zero and infinity. Goulden, Jackson and Vakil have conjectured that double Hurwitz numbers also arise as intersection numbers on moduli spaces.In this paper, we repackage double Hurwitz numbers as enumerations of branched covers weighted by certain monomials and conjecture that they are governed by the topological recursion. Evidence is provided in the form of the associated quantum curve and low genus calculations. We furthermore reduce the conjecture to a weaker one, concerning a certain polynomial structure of double Hurwitz numbers. Via the topological recursion framework, our main conjecture should lead to a direct connection to enumerative geometry, thus shedding light on the aforementioned conjecture of Goulden, Jackson and Vakil.

show abstract

Think globally, compute locally

Cited by 89 publications

References 51 publications

Weighted Hurwitz Numbers and Topological Recursion

Weighted Hurwitz Numbers and Topological Recursion

Schwinger–Dyson and loop equations for a product of square Ginibre random matrices

Towards the topological recursion for double Hurwitz numbers

Contact Info

Product

Resources

About