Grothendieck's Dessins Research Articles

Explicit formulas for a family of hypermaps beyond the one-face case

Enumeration of hypermaps (or Grothendieck's dessins d'enfants) is widely studied in many fields. In particular, enumerating hypermaps with a fixed edge-type according to the number of faces and genus is one topic of great interest. The first systematic study of hypermaps with one face and any fixed edge-type is the work of Jackson (1987) [23] using group characters. Stanley later (2011) obtained the genus distribution polynomial of one-face hypermaps of any fixed edge-type expressed in terms of the backward shift operator. There is also enormous amount of work on enumerating one-face hypermaps of specific edge-types. Hypermaps with more faces are generally much harder to enumerate and results are rare. Our main results here are formulas for the genus distribution polynomials for a family of typical two-face hypermaps including almost all edge-types, the purely imaginary zeros property of these polynomials, and the log-concavity of the coefficients.

Triply mixed coverings of arbitrary base curves: quasimodularity, quantum curves and a mysterious topological recursion

Simple Hurwitz numbers enumerate branched morphisms between Riemann surfaces with fixed ramification data. In recent years, several variants of this notion for genus $0$ base curves have appeared in the literature. Among them are so-called monotone Hurwitz numbers, which are related to the HCIZ integral in random matrix theory and strictly monotone Hurwitz numbers which count certain Grothendieck dessins d'enfants. We generalise the notion of Hurwitz numbers to interpolations between simple, monotone and strictly monotone Hurwitz numbers to any genus and any number of arbitrary but fixed ramification profiles. This yields generalisations of several results known for Hurwitz numbers. When the target surface is of genus one, we show that the generating series of these interpolated Hurwitz numbers are quasimodular forms. In the case that all ramification is simple, we refine this result by writing this series as a sum of quasimodular forms corresonding to tropical covers weighted by Gromov-Witten invariants. Moreover, we derive a quantum curve for monotone and Grothendieck dessins d'enfants Hurwitz numbers for arbitrary genera and one arbitrary but fixed ramification profile. Thus, we obtain spectral curves via the semiclassical limit as input data for the CEO topological recursion. Astonishingly, we find that the CEO topological recursion for the genus $1$ spectral curve of the strictly monotone Hurwitz numbers compute the monotone Hurwitz numbers in genus $0$. Thus, we give a new proof that monotone Hurwitz numbers satisfy CEO topological recursion. This points to an unknown relation between those enumerants. Finally, specializing to target surface $\mathbb{P}^1$, we find recursions for monotone and Grothendieck dessins d'enfants double Hurwitz numbers, which enables the computation of the respective Hurwitz numbers for any genera with one arbitrary but fixed ramification profile.

Quantum Statistical Mechanics of the Absolute Galois Group

We present possible extensions of the quantum statistical mechanical formulation of class field theory to the non-abelian case, based on the action of the absolute Galois group on Grothendieck's dessins d'enfant, the embedding in the Grothendieck-Teichm\"uller group, and the Drinfeld-Ihara involution.

Wall-crossing and recursion formulae for tropical Jucys covers

Hurwitz numbers enumerate branched genus g g covers of the Riemann sphere with fixed ramification data or equivalently certain factorisations of permutations. Double Hurwitz numbers are an important class of Hurwitz numbers, obtained by considering ramification data with a specific structure. They exhibit many fascinating properties, such as a beautiful piecewise polynomial structure, which has been well-studied in the last 15 years. In particular, using methods from tropical geometry, it was possible to derive wall-crossing formulae for double Hurwitz numbers in arbitrary genus. Further, double Hurwitz numbers satisfy an explicit recursive formula. In recent years several related enumerations have appeared in the literature. In this work, we focus on two of those invariants, so-called monotone and strictly monotone double Hurwitz numbers. Monotone double Hurwitz numbers originate from random matrix theory, as they appear as the coefficients in the asymptotic expansion of the famous Harish-Chandra–Itzykson–Zuber integral. Strictly monotone double Hurwitz numbers are known to be equivalent to an enumeration of Grothendieck dessins d’enfants. These new invariants share many structural properties with double Hurwitz numbers, such as piecewise polynomiality. In this work, we enlarge upon this study and derive new explicit wall-crossing and recursive formulae for monotone and strictly monotone double Hurwitz numbers. The key ingredient is a new interpretation of monotone and strictly monotone double Hurwitz numbers in terms of tropical covers, which was recently derived by the authors. An interesting observation is the fact that monotone and strictly monotone double Hurwitz numbers satisfy wall-crossing formulae, which are almost identical to the classical double Hurwitz numbers.

Counting surface branched covers

Abstract To a branched cover f between orientable surfaces one can associate a certain branch datum, that encodes the combinatorics of the cover. This satisfies a compatibility condition called the Riemann-Hurwitz relation. The old but still partly unsolved Hurwitz problem asks whether for a given abstract compatible branch datum there exists a branched cover f such that . One can actually refine this problem and ask how many these f's exist, but one must of course decide what restrictions one puts on such f’s, and choose an equivalence relation up to which one regards them. As it turns out, quite a few natural choices for this relation are possible. In this short note we carefully analyze all these choices and show that the number of actually distinct ones is only three. To see that these three choices are indeed different from each other we employ Grothendieck's dessins d'enfant.

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

Hurwitz numbers count genus g, degree d covers of the complex projective line with fixed branched locus and fixed ramification data. An equivalent description is given by factorisations in the symmetric group. Simple double Hurwitz numbers are a class of Hurwitz-type counts of specific interest. In recent years a related counting problem in the context of random matrix theory was introduced as so-called monotone Hurwitz numbers. These can be viewed as a desymmetrised version of the Hurwitz-problem. A combinatorial interpolation between simple and monotone double Hurwitz numbers was introduced as mixed double Hurwitz numbers and it was proved that these objects are piecewise polynomial in a certain sense. Moreover, the notion of strictly monotone Hurwitz numbers has risen in interest as it is equivalent to a certain Grothendieck dessins d’enfant count. In this paper, we introduce a combinatorial interpolation between simple, monotone and strictly monotone double Hurwitz numbers as triply interpolated Hurwitz numbers. Our aim is twofold: using a connection between triply interpolated Hurwitz numbers and tropical covers in terms of so-called monodromy graphs, we give algorithms to compute the polynomials for triply interpolated Hurwitz numbers in all genera using Erhart theory. We further use this approach to study the wall-crossing behaviour of triply interpolated Hurwitz numbers in genus 0 in terms of related Hurwitz-type counts. All those results specialise to the extremal cases of simple, monotone and Grothendieck dessins d’enfants Hurwitz numbers.

Quasi-Polynomiality of Monotone Orbifold Hurwitz Numbers and Grothendieck's Dessins d'Enfants

We prove quasi-polynomiality for monotone and strictly monotone orbifold Hurwitz numbers. The second enumerative problem is also known as enumeration of a special kind of Grothendieck's dessins d'enfants or r -hypermaps. These statements answer positively two conjectures proposed by Do-Karev and Do-Manescu. We also apply the same method to the usual orbifold Hurwitz numbers and obtain a new proof of the quasi-polynomiality in this case. In the second part of the paper we show that the property of quasi-polynomiality is equivalent in all these three cases to the property that the n -point generating function has a natural representation on the n -th cartesian powers of a certain algebraic curve. These representations are necessary conditions for the Chekhov-Eynard-Orantin topological recursion.

Wall-crossing formulae and strong piecewise polynomiality for mixed Grothendieck dessins d'enfant, monotone, and double simple Hurwitz numbers

We derive explicit formulae for the generating series of mixed Grothendieck dessins d'enfant/monotone/simple Hurwitz numbers, via the semi-infinite wedge formalism. This reveals the strong piecewise polynomiality in the sense of Goulden–Jackson–Vakil, generalising a result of Johnson, and provides a new explicit proof of the piecewise polynomiality of the mixed case. Moreover, we derive wall-crossing formulae for the mixed case. These statements specialise to any of the three types of Hurwitz numbers, and to the mixed case of any pair.

On Mono-nodal Trees and Genus One Dessins of Pakovich-Zapponi Type

In this paper, we classify Grothendieck dessins of X-shaped plane trees defined over the rationals.

A sharp bound on the number of real intersection points of a sparse plane curve with a line

We prove that the number of real intersection points of a real line with a real plane curve defined by a polynomial with at most t monomials is either infinite or does not exceed 6t−7. This improves a result by M. Avendaño. Furthermore, we prove that this bound is sharp for t=3 with the help of Grothendieck's dessins d'enfant.

Three-point Lie algebras and Grothendieck’s dessins d’enfants

We define and classify the analogues of the affine Kac-Moody Lie algebras for the ring corresponding to the complex projective line minus three points. The classification is given in terms of Grothendieck's dessins d'enfants. We also study the question of conjugacy of Cartan subalgebras for these algebras.

Hecke Groups, Dessins d'Enfants, and the Archimedean Solids

Grothendieck's dessins d'enfants arise with ever-increasing frequency in many areas of 21st century mathematical physics. In this paper, we review the connections between dessins and the theory of Hecke groups. Focussing on the restricted class of highly symmetric dessins corresponding to the so-called Archimedean solids, we apply this theory in order to provide a means of computing representatives of the associated conjugacy classes of Hecke subgroups in each case. The aim of this paper is to demonstrate that dessins arising in mathematical physics can point to new and hitherto unexpected directions for further research. In addition, given the particular ubiquity of many of the dessins corresponding to the Archimedean solids, the hope is that the computational results of this paper will prove useful in the further study of these objects in mathematical physics contexts.

Dessins d’enfants in N = 2 $$ \mathcal{N}=2 $$ generalised quiver theories

We study Grothendieck's dessins d'enfants in the context of the $\mathcal{N}=2$ supersymmetric gauge theories in $\left(3+1\right)$ dimensions with product $SU\left(2\right)$ gauge groups which have recently been considered by Gaiotto et al. We identify the precise context in which dessins arise in these theories: they are the so-called ribbon graphs of such theories at certain isolated points in the Coulomb branch of the moduli space. With this point in mind, we highlight connections to other work on trivalent dessins, gauge theories, and the modular group.

Quantum contextual finite geometries from dessins d'enfants

We point out an explicit connection between graphs drawn on compact Riemann surfaces defined over the field [Formula: see text] of algebraic numbers — the so-called Grothendieck's dessins d'enfants — and a wealth of distinguished point-line configurations. These include simplices, cross-polytopes, several notable projective configurations, a number of multipartite graphs and some "exotic" geometries. Among them, remarkably, we find not only those underlying Mermin's magic square and magic pentagram, but also those related to the geometry of two- and three-qubit Pauli groups. Of particular interest is the occurrence of all the three types of slim generalized quadrangles, namely GQ(2, 1), GQ(2, 2) and GQ(2, 4), and a couple of closely related graphs, namely the Schläfli and Clebsch ones. These findings seem to indicate that dessins d'enfants may provide us with a new powerful tool for gaining deeper insight into the nature of finite-dimensional Hilbert spaces and their associated groups, with a special emphasis on contextuality.

Geometry of contextuality from Grothendieck’s coset space

The geometry of cosets in the subgroups H of the two-generator free group G =\textless{} a, b \textgreater{} nicely fits, via Grothendieck's dessins d'enfants, the geometry of commutation for quantum observables. Dessins stabilize point-line incidence geometries that reflect the commutation of (generalized) Pauli operators [Information 5, 209 (2014); 1310.4267 and 1404.6986 (quant-ph)]. Now we find that the non-existence of a dessin for which the commutator (a, b) = a^ (--1) b^( --1) ab precisely corresponds to the commutator of quantum observables [A, B] = AB -- BA on all lines of the geometry is a signature of quantum contextuality. This occurs first at index |G : H| = 9 in Mermin's square and at index 10 in Mermin's pentagram, as expected. Commuting sets of n-qubit observables with n \textgreater{} 3 are found to be contextual as well as most generalized polygons. A geometrical contextuality measure is introduced.

Enumeration of Grothendieck's Dessins and KP Hierarchy

Branched covers of the complex projective line ramified over |$0,1,$| and |$\infty $| (Grothendieck's dessins d'enfant) of fixed genus and degree are effectively enumerated. More precisely, branched covers of a given ramification profile over |$\infty $| and given numbers of preimages of |$0$| and |$1$| are considered. The generating function for the numbers of such covers is shown to satisfy a partial differential equation (PDE) that determines it uniquely modulo a simple initial condition. Moreover, this generating function satisfies an infinite system of PDE's called the Kadomtsev–Petviashvili (KP) hierarchy. A specification of this generating function for certain values of parameters generates the numbers of dessins of given genus and degree, thus providing a fast algorithm for computing these numbers.

Real trigonal curves and real elliptic surfaces of type I

We study real trigonal curves and elliptic surfaces of type I (over a base of an arbitrary genus) and their fiberwise equivariant deformations. The principal tool is a real version of Grothendieck's dessins d'enfants. We give a description of maximally inflected trigonal curves of type I in terms of the combinatorics of sufficiently simple graphs and, in the case of the rational base, obtain a complete classification of such curves. As a consequence, these results lead to conclusions concerning real Jacobian elliptic surfaces of type I with all singular fibers real.

True trees are dense

We show that any compact, connected set $K$ in the plane can be approximated by the critical points of a polynomial with two critical values. Equivalently, $K$ can be approximated in the Hausdorff metric by a true tree in the sense of Grothendieck's dessins d'enfants.

Plane sextics with a type $\bold{E}_8$ singular point

We construct explicit geometric models for and compute the fundamen- tal groups of all plane sextics with simple singularities only and with at least one type E8 singular point. In particular, we discover four new sextics with nonabelian fundamental groups; two of them are irreducible. The groups of the two irreducible sextics found are finite. The principal tool used is the reduction to trigonal curves and Grothendieck's dessins d'enfants. 1.1. Principal results. This paper is a continuation of my paper (6), where we started the study of the equisingular deformation families and the fundamental groups of plane sextics (i.e., curves B ⊂ P 2 of degree six) with a distinguished triple singular point, using the representation of such sextics via trigonal curves in Hirzebruch surfaces and Grothendieck's dessins d'enfants. (All varieties in the paper are over C and are considered in their Hausdorff topology.) Recall that, in spite of the fact that the deformation classification of sextics can be reduced to a relatively simple, although tedious, arithmetical problem, see (2), the geometry of the pairs (P 2 , B) remains a terra incognita, as the construction relies upon the global Torelli theorem for K3-surfaces and is quite implicit. On the contrary, the approach suggested in (6), although not resulting in a defining equation for B, gives one a fairly good understanding of the topology of (P 2 , B); in particular, it is sufficient for the computation of the fundamental groupπ1(P 2 r B). A few other applications of this approach and more motivation can be found in (6); for a brief overview of the latest achievements on the subject, see C. Eyral, M. Oka (7). In the present paper, we deal with the case when the distinguished triple point in question is of type E8. (The case of a type E7 singular point was considered in (6), and the case of E6 is the subject of a forthcoming paper.) As in (6), a simple trick with the skeletons reduces most sextics B ⊂ P 2 to certain trigonal curves ¯ Bin �2 (instead of the original surface �3); this simplifies dramatically the classification of the sextics and the computation of their fundamental groups. It is still unclear if there is a simple geometric relation between B and ¯ B ' . Throughout the paper, we consider a plane sextic B ⊂ P 2 satisfying the following conditions: (∗) B has simple singularities only, and B has a distinguished singular point P of type E8.

Equations determining Belyi pairs, with applications to anti-Vandermonde systems

This paper deals with Grothendieck dessins d'enfants, i.e., tamely embedded graphs on surfaces, and Belyi pairs, i.e., rational functions with at most three critical values on algebraic curves. The relationship between these objects was promoted by Grothendieck. We investigate combinatorics of systems of equations determining a Belyi pair corresponding to a given dessin. Some properties of extra, or so-called parasitic, solutions of such systems are described. As a corollary, we obtain some applications concerning anti-Vandermonde systems.