Weighted Hurwitz Numbers and Topological Recursion

Abstract

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...

Matrix model generating function for quantum weighted Hurwitz numbers

“…As detailed in [4][5][6], τ (Hq,β) (t) is the KP τ -function corresponding to the Grassmannian element W (Hq,β) spanned by the basis elements {φ If τ (Hq,β) (t) is evaluated at the trace invariants…”

Generating weighted Hurwitz numbers

“…Finally, we quote the following further result from [3,5], which expresses the connected multicurrent correlators in terms of pair correlators.…”

“…It is well-known that KP τ -functions of hypergeometric type may serve as combinatorial generating functions for weighted Hurwitz numbers [28,30,6,15,21,16,18]. An efficient way of computing the latter is to make use of the associated multicurrent correlators [4,5]. This method is implemented in the following, without recourse to matrix integral representations [1,2,9] or topological recursion [10,8,25,3,4,5].…”

“…An efficient way of computing the latter is to make use of the associated multicurrent correlators [4,5]. This method is implemented in the following, without recourse to matrix integral representations [1,2,9] or topological recursion [10,8,25,3,4,5]. Section 2 reviews the use of KP and 2D-Toda τ -functions as generating functions for weighted Hurwitz numbers.…”

“…. , x n ) introduced in [4,5] in the context of the topological recursion (TR) approach. As explained in Proposition 2.2, these provide generating functions for single weighted Hurwitz numbers H d G (µ), grouped according to the length n = ℓ(µ) of the partition µ giving the ramification profile over a selected 2 Generating functions for weighted Hurwitz numbers 2…”