Quantum curves for the enumeration of ribbon graphs and hypermaps

Abstract

Abstract. The topological recursion of Eynard and Orantin governs a variety of problems in enumerative geometry and mathematical physics. The recursion uses the data of a spectral curve to define an infinite family of multidifferentials. It has been conjectured that, under certain conditions, the spectral curve possesses a non-commutative quantisation whose associated differential operator annihilates the partition function for the spectral curve. In this paper, we determine the quantum curves and partition functions for an infinite sequence of enumerative problems involving generalisations of ribbon graphs known as hypermaps. These results give rise to an explicit conjecture relating hypermap enumeration to the topological recursion and we provide evidence to support this conjecture.

Reconstructing WKB from topological recursion

“…More precisely, the claim is that there exists a Schrödinger-like ordinary differential operator, which is a quantization of the original spectral curve (which is why it is called a quantum curve), and whose WKB asymptotic solution is reconstructed by the topological recursion applied to this spectral curve. This claim [31,8,9,10] has been verified for a small number of genus zero spectral curves, in various algebro-geometric contexts [1,8,14,21,22,23,25,24,26,47,52,53,56]. In the context of knot theory, this claim provides a constructive approach to the well known AJ-conjecture [38], which has been studied in a number of papers [20,11,20,36,37,39,42].…”

“…, a. The case a = 2 arises in the study of enumeration of ribbon graphs [53,54], while the a > 2 case arises in the enumeration of a-hypermaps [22,28]. 6.2.1. a = 2.…”

Counting curves on surfaces

“…However, they are not sufficient. For instance, N 7 3,2 (2n + 1, 1) = N 3,2,n+2 (2n + 1, 1) = 0, but max(0, 6 − n) = max(k, 2g + n − 1 − 1 2 b i ) ≤ 7 = t ≤ min(3g − 3 + n, 2g + n − 1) = min (8,7). To see why, suppose there were such an arc diagram; then there must be a (necessarily non-separating) arc connecting the two boundary components.…”

“…There has been a great deal of recent work demonstrating that many enumerative problems formulated in terms of surfaces display similar phenomena: polynomiality, recursion, and differential forms and generating functions obeying physically suggestive equations. Such problems arise, for instance, in matrix models [5], the theory of Hurwitz numbers [1,3,6,17], moduli spaces of curves [8,28,31,32], Gromov-Witten theory [2,13,16,19,33] and combinatorics [7,12,14,29].…”

A Monodromy Graph Approach to the Piecewise Polynomiality of Simple, Monotone and Grothendieck Dessins d’enfants Double Hurwitz Numbers

“…Further, the notion of strictly monotone Hurwitz numbers have gained attention as is equivalent to counting certain Grothendieck dessins d'enfants [ALS16]. As for monotone Hurwitz numbers, topological recursion was proved for many cases of strictly monotone double Hurwitz numbers [Nor09,DM13,DMSS13,KZ15]. A combinatorial interpolation between double Hurwitz numbers and monotone double Hurwitz numbers was studied in [GGN16] as mixed Hurwitz numbers.…”