A matrix model for simple Hurwitz numbers, and topological recursion

Abstract

Classical Hurwitz numbers count branched covers of the Riemann sphere with prescribed ramification data, or equivalently, factorisations in the symmetric group with prescribed cycle structure data. Monotone Hurwitz numbers restrict the enumeration by imposing a further monotonicity condition on such factorisations. In this paper, we prove that monotone Hurwitz numbers arise from the topological recursion of Eynard and Orantin applied to a particular spectral curve. We furthermore derive a quantum curve for monotone Hurwitz numbers. These results extend the collection of enumerative problems known to be governed by the paradigm of topological recursion and quantum curves, as well as the list of analogues between monotone Hurwitz numbers and their classical counterparts.

2D Toda τ-Functions as Combinatorial Generating Functions

“…satisfy the infinite set of Hirota bilinear equations defining the KP hierarchy, or some reduction thereof. These include the Kontsevich matrix integral [29], which is a KdV τ -function, the generator for Hodge invariants [27], the matrix integrals that generate single Hurwitz numbers [4,10,32,37], and the ones for Belyi curves and dessins d'enfants [2,28,41]. Other generating functions are known to be τ -functions of the 2D Toda hierarchy, some of which are also representable as matrix integrals.…”

Matrix models for random partitions

“…A huge number of interesting topics, such as the phase transitions of the obtained matrix integrals [24], their role in M-theory of matrix models (decomposition formulas) [37] and Virasoro-type constraints [21,38] (in particular, an application of the powerful Eynard technique [22,25,26]) for them are beyond the scope of this letter. It is not obvious to us if there exists an infinite set of the Virasoro-type constraints that is an algebra of the low-order differential operators, which act in the space of the time variables t k ,t k and, probably, s k , and annihilate the propagator P N (t,t; s).…”

“…Another important example is the generating function of the simple Hurwitz numbers. Here two different matrix integrals are known: one [26] developing the ideas of [25], and another [27] with usual integration contours but non-flat measure. These two matrix models are related through the Fourier-Laplace transform [28].…”

Unitary integrals and related matrix models

“…One of the basic intrigues in it is connected with the interplay between groups of two different types: linear (GL) and symmetric (S). This interplay is now attracting increasing attention because of the close link found between, on one hand, Yang-Mills theory, best represented in the matrix theory context by the - [33], and, on the other hand, the Hurwitz theory of ramified coverings of Riemann surfaces [34]- [40], represented by the Hurwitz-Kontsevich partition function [35]- [38] and rather unusual matrix models [36], [37]. U = e iH =⇒ δU 2 = tr(U † δU U † δU ) = tr(δH) 2 =⇒ [dU ] = dH,…”