arXiv subjects

Norman Do

Publications and source records attributed to Norman Do.

At least 19 recordsLinked to original sources

arXiv2025

From double-scaled SYK correlators to Weil-Petersson volumes

Okuyama introduced a family of polynomials, whose coefficients depend on a parameter $q$, in his study of correlators in the double-scaled SYK model. He verified in small cases that their coefficients can be expressed in terms of certain $q$-zeta values and that the polynomials recover the Weil-Petersson volumes of moduli spaces studied by Mirzakhani under a certain $q \to 1$ limit. In this paper, we provide mathematically rigorous proofs of these two phenomena. The authors previously defined natural $q$-deformations of the Weil-Petersson volumes of moduli spaces of curves. We prove that these polynomials appear as the top degree part of Okuyama's polynomials. Our work provides a link between the two topics of the title, which hints at a ``quantum'' Weil-Petersson geometry and a combinatorial-geometric approach to double-scaled SYK correlators.

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arXiv2025

A q-analogue of Mirzakhani's recursion for Weil-Petersson volumes

We define q-analogues of Mirzakhani's recursion for Weil-Petersson volumes and the Stanford-Witten recursion for super Weil-Petersson volumes. Okuyama recently introduced a q-deformation of the Gaussian Hermitian matrix model which produces quasi-polynomials that recover the Weil-Petersson volumes via a rescaled q to 1 limit. The q-deformations of the Weil-Petersson volumes produced here agree with the top degree terms of Okuyama's quasi-polynomials and suggest a variation of Okuyama's methods to the super setting.

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arXiv2025

From Weingarten calculus for real Grassmannians to deformations of monotone Hurwitz numbers and Jucys-Murphy elements

The present work is inspired by three interrelated themes: Weingarten calculus for integration over unitary groups, monotone Hurwitz numbers which enumerate certain factorisations of permutations into transpositions, and Jucys-Murphy elements in the symmetric group algebra. The authors and Moskovsky recently extended this picture to integration on complex Grassmannians, leading to a deformation of the monotone Hurwitz numbers to polynomials that are conjectured to satisfy remarkable interlacing phenomena. In this paper, we consider integration on the real Grassmannian $\mathrm{Gr}_\mathbb{R}(M,N)$, interpreted as the space of $N \times N$ idempotent real symmetric matrices of rank $M$. We show that in the regime of large $N$ and fixed $\frac{M}{N}$, such integrals have expansions whose coefficients are variants of monotone Hurwitz numbers that are polynomials in the parameter $t = 1 - \frac{N}{M}$. We define a "$b$-Weingarten calculus", without reference to underlying matrix integrals, that recovers the unitary case at $b = 0$ and the orthogonal case at $b = 1$. The $b$-monotone Hurwitz numbers, previously introduced by Bonzom, Chapuy and Dolega, arise naturally in this context as monotone factorisations of pair partitions. The $b$- and $t$-deformations can be combined to form a common generalisation, leading to the notion of $bt$-monotone Hurwitz numbers, for which we state several results and conjectures. Finally, we introduce certain linear operators inspired by the aforementioned $b$-Weingarten calculus that can be considered as $b$-deformations of the Jucys-Murphy elements in the symmetric group algebra. We make several conjectures regarding these operators that generalise known properties of the Jucys-Murphy elements and make a connection to the family of Jack symmetric functions.

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arXiv2024

The structure of Hurwitz numbers with fixed ramification profile and varying genus

In 1891, Hurwitz introduced the enumeration of genus $g$, degree $d$, branched covers of the Riemann sphere with simple ramification over prescribed points and no branching elsewhere. He showed that for fixed degree $d$, the enumeration possesses a remarkable structure. More precisely, it can be expressed as a linear combination of exponentials $m^{2g-2+2d}$, where $m$ ranges over the integers from $1$ to $\binom{d}{2}$. In this paper, we generalise this structural result to Hurwitz numbers that enumerate branched covers which also have a prescribed ramification profile over one point. Our proof fundamentally uses the infinite wedge space, in particular the connected correlators of products of $\mathcal{E}$-operators. The recent study of Hurwitz numbers has often focussed on their structure with fixed genus and varying ramification profile. Our main result is orthogonal to this, allowing for the explicit calculation and the asymptotic analysis of Hurwitz numbers in large genus. We pose the broad question of which other enumerative problems exhibit analogous structure. We prove that orbifold Hurwitz numbers can also be expressed as a linear combination of exponentials and conjecture that monotone Hurwitz numbers share a similar structure, but with the inclusion of an additional linear term.

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arXiv2024

Volume bounds for hyperbolic rod complements in the 3-torus

The study of rod complements is motivated by rod packing structures in crystallography. We view them as complements of links comprised of Euclidean geodesics in the 3-torus. Recent work of the second author classifies when such rod complements admit hyperbolic structures, but their geometric properties are yet to be well understood. In this paper, we provide upper and lower bounds for the volumes of all hyperbolic rod complements in terms of rod parameters, and show that these bounds may be loose in general. We introduce better and asymptotically sharp volume bounds for a family of rod complements. The bounds depend only on the lengths of the continued fractions formed from the rod parameters.

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arXiv2023

Integration on complex Grassmannians, deformed monotone Hurwitz numbers, and interlacing phenomena

We introduce a family of polynomials, which arise in three distinct ways: in the large $N$ expansion of a matrix integral, as a weighted enumeration of factorisations of permutations, and via the topological recursion. More explicitly, we interpret the complex Grassmannian $\mathrm{Gr}(M,N)$ as the space of $N \times N$ idempotent Hermitian matrices of rank $M$ and develop a Weingarten calculus to integrate products of matrix elements over it. In the regime of large $N$ and fixed ratio $\frac{M}{N}$, such integrals have expansions whose coefficients count factorisations of permutations into monotone sequences of transpositions, with each sequence weighted by a monomial in $t = 1 - \frac{N}{M}$. This gives rise to the desired polynomials, which specialise to the monotone Hurwitz numbers when $t = 1$. These so-called deformed monotone Hurwitz numbers satisfy a cut-and-join recursion, a one-point recursion, and the topological recursion. Furthermore, we conjecture on the basis of overwhelming empirical evidence that the deformed monotone Hurwitz numbers are real-rooted polynomials whose roots satisfy remarkable interlacing phenomena. An outcome of our work is the viewpoint that the topological recursion can be used to "topologise" sequences of polynomials, and we claim that the resulting families of polynomials may possess interesting properties. As a further case study, we consider a weighted enumeration of dessins d'enfant and conjecture that the resulting polynomials are also real-rooted and satisfy analogous interlacing properties.

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arXiv2022

The tropological vertex

The theory of the topological vertex was originally proposed by Aganagic, Klemm, Mari\~no and Vafa as a means to calculate open Gromov-Witten invariants of toric Calabi-Yau threefolds. In this paper, we place the topological vertex within the context of relative Gromov-Witten invariants of log Calabi-Yau manifolds and describe how these invariants can be effectively computed via a gluing formula for the enumeration of tropical curves in a singular integral affine space. This richer context allows us to prove that the topological vertex possesses certain tropical symmetries. These symmetries are captured by the action of a quantum torus Lie algebra that is related to a quantisation of the Lie algebra of the tropical vertex group of Gross, Pandharipande and Siebert. Finally, we demonstrate how this algebra of symmetries leads to an explicit description of the topological vertex and related Gromov-Witten invariants.

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arXiv2020

On the Goulden-Jackson-Vakil conjecture for double Hurwitz numbers

Goulden, Jackson and Vakil observed a polynomial structure underlying one-part double Hurwitz numbers, which enumerate branched covers of $\mathbb{CP}^1$ with prescribed ramification profile over $\infty$, a unique preimage over 0, and simple branching elsewhere. This led them to conjecture the existence of moduli spaces and tautological classes whose intersection theory produces an analogue of the celebrated ELSV formula for single Hurwitz numbers. In this paper, we present three formulas that express one-part double Hurwitz numbers as intersection numbers on certain moduli spaces. The first involves Hodge classes on moduli spaces of stable maps to classifying spaces; the second involves Chiodo classes on moduli spaces of spin curves; and the third involves tautological classes on moduli spaces of stable curves. We proceed to discuss the merits of these formulas against a list of desired properties enunciated by Goulden, Jackson and Vakil. Our formulas lead to non-trivial relations between tautological intersection numbers on moduli spaces of stable curves and hints at further structure underlying Chiodo classes. The paper concludes with generalisations of our results to the context of spin Hurwitz numbers.

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arXiv2020

Double Hurwitz numbers: polynomiality, topological recursion and intersection theory

Double Hurwitz numbers enumerate branched covers of $\mathbb{CP}^1$ with prescribed ramification over two points and simple ramification elsewhere. In contrast to the single case, their underlying geometry is not well understood. In previous work by the second- and third-named authors, the double Hurwitz numbers were conjectured to satisfy a polynomiality structure and to be governed by the topological recursion, analogous to existing results concerning single Hurwitz numbers. In this paper, we resolve these conjectures by a careful analysis of the semi-infinite wedge representation for double Hurwitz numbers, by pushing further methods previously used for other Hurwitz problems. We deduce a preliminary version of an ELSV-like formula for double Hurwitz numbers, by deforming the Johnson-Pandharipande-Tseng formula for orbifold Hurwitz numbers and using properties of the topological recursion under variation of spectral curves. In the course of this analysis, we unveil certain vanishing properties of the Chiodo classes.

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arXiv2019

Counting non-crossing permutations on surfaces of any genus

Given a surface with boundary and some points on its boundary, a polygon diagram is a way to connect those points as vertices of non-overlapping polygons on the surface. Such polygon diagrams represent non-crossing permutations on a surface with any genus and number of boundary components. If only bigons are allowed, then it becomes an arc diagram. The count of arc diagrams is known to have a rich structure. We show that the count of polygon diagrams exhibits the same interesting behaviours, in particular it is almost polynomial in the number of points on the boundary components, and the leading coefficients of those polynomials are the intersection numbers on the compactified moduli space of curves.

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arXiv2019

Local topological recursion governs the enumeration of lattice points in $\overline{\mathcal M}_{g,n}$

The second author and Norbury initiated the enumeration of lattice points in the Deligne-Mumford compactifications of moduli spaces of curves. They showed that the enumeration may be expressed in terms of polynomials, whose top and bottom degree coefficients store psi-class intersection numbers and orbifold Euler characteristics of $\overline{\mathcal M}_{g,n}$, respectively. Furthermore, they ask whether the enumeration is governed by the topological recursion and whether the intermediate coefficients also store algebro-geometric information. In this paper, we prove that the enumeration does indeed satisfy the topological recursion, although with a modification to the initial spectral curve data. Thus, one can consider this to be one of the first known instances of a natural enumerative problem governed by the so-called local topological recursion. Combining the present work with the known relation between local topological recursion and cohomological field theory should uncover the geometric meaning of the intermediate coefficients of the aforementioned polynomials.

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arXiv2019

Relating ordinary and fully simple maps via monotone Hurwitz numbers

A direct relation between the enumeration of ordinary maps and that of fully simple maps first appeared in the work of the first and last authors. The relation is via monotone Hurwitz numbers and was originally proved using Weingarten calculus for matrix integrals. The goal of this paper is to present two independent proofs that are purely combinatorial and generalise in various directions, such as to the setting of stuffed maps and hypermaps. The main motivation to understand the relation between ordinary and fully simple maps is the fact that it could shed light on fundamental, yet still not well-understood, problems in free probability and topological recursion.

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arXiv2018

Generalisations of the Harer-Zagier recursion for 1-point functions

Harer and Zagier proved a recursion to enumerate gluings of a $2d$-gon that result in an orientable genus $g$ surface, in their work on Euler characteristics of moduli spaces of curves. Analogous results have been discovered for other enumerative problems, so it is natural to pose the following question: how large is the family of problems for which these so-called 1-point recursions exist? In this paper, we prove the existence of 1-point recursions for a class of enumerative problems that have Schur function expansions. In particular, we recover the Harer-Zagier recursion, but our methodology also applies to the enumeration of dessins d'enfant, to Bousquet-Mélou-Schaeffer numbers, to monotone Hurwitz numbers, and more. On the other hand, we prove that there is no 1-point recursion that governs simple Hurwitz numbers. Our results are effective in the sense that one can explicitly compute particular instances of 1-point recursions, and we provide several examples. We conclude the paper with a brief discussion and a conjecture relating 1-point recursions to the theory of topological recursion.

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arXiv2018

Towards the topological recursion for double Hurwitz numbers

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.

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arXiv2017

The completeness of the Bethe ansatz for the periodic ASEP

The asymmetric simple exclusion process (ASEP) for N particles on a ring with L sites may be analyzed using the Bethe ansatz. In this paper, we provide a rigorous proof that the Bethe ansatz is complete for the periodic ASEP. More precisely, we show that for all but finitely many values of the hopping rate, the solutions of the Bethe ansatz equations do indeed yield all L choose N eigenstates. The proof follows ideas of Langlands and Saint-Aubin, which draw upon a range of techniques from algebraic geometry, topology and enumerative combinatorics.

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arXiv2016

Topological recursion on the Bessel curve

The Witten-Kontsevich theorem states that a certain generating function for intersection numbers on the moduli space of stable curves is a tau-function for the KdV integrable hierarchy. This generating function can be recovered via the topological recursion applied to the Airy curve $x=\frac{1}{2}y^2$. In this paper, we consider the topological recursion applied to the irregular spectral curve $xy^2=\frac{1}{2}$, which we call the Bessel curve. We prove that the associated partition function is also a KdV tau-function, which satisfies Virasoro constraints, a cut-and-join type recursion, and a quantum curve equation. Together, the Airy and Bessel curves govern the local behaviour of all spectral curves with simple branch points.

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arXiv2016

Counting curves on surfaces

In this paper we consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology. Consider a real 2-dimensional compact surface $S$, and fix a number of points $F$ on its boundary. We ask: how many configurations of disjoint arcs are there on $S$ whose boundary is $F$? We find that this enumerative problem, counting curves on surfaces, has a rich structure. For instance, we show that the curve counts obey an effective recursion, in the general framework of topological recursion. Moreover, they exhibit quasi-polynomial behaviour. This "elementary curve-counting" is in fact related to a more advanced notion of "curve-counting" from algebraic geometry or symplectic geometry. The asymptotics of this enumerative problem are closely related to the asymptotics of volumes of moduli spaces of curves, and the quasi-polynomials governing the enumerative problem encode intersection numbers on moduli spaces. Furthermore, among several other results, we show that generating functions and differential forms for these curve counts exhibit structure that is reminiscent of the mathematical physics of free energies, partition functions, topological recursion, and quantum curves.

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arXiv2015

Monotone orbifold Hurwitz numbers

In general, 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. In this paper, we initiate the study of monotone orbifold Hurwitz numbers. These are simultaneously variations of the orbifold case and generalisations of the monotone case, both of which have been previously studied in the literature. We derive a cut-and-join recursion for monotone orbifold Hurwitz numbers, determine a quantum curve governing their wave function, and state an explicit conjecture relating them to topological recursion.

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