arXiv subjects

Paul Norbury

Publications and source records attributed to Paul Norbury.

At least 19 recordsLinked to original sources

arXiv2026

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.

math.AG↗

arXiv2026

Spin structures and measures on the moduli space of curves

This article arose out of notes for a CIME summer school mini-course in Cetraro. In it we develop differential-geometric constructions on the moduli space ${\cal M}_{g,n}^{\rm spin}$ of smooth spin curves, with particular emphasis on their interpretation in hyperbolic geometry. We relate the locally constant sheaf arising from a hyperbolic spin structure to the corresponding holomorphic sheaf, extending this known relationship to spin curves with Ramond marked points. We use the hyperbolic description to construct characteristic forms and spin measures on moduli spaces of hyperbolic surfaces with geodesic boundary, providing a framework for applying Teichmüller-theoretic methods to their volumes and volume recursions. Finally, we study separately the even and odd components of spin moduli space and find symmetries between their volume contributions which are not apparent from the contributions of individual boundary strata.

math.AG↗

arXiv2025

Enumerative geometry via the moduli space of super Riemann surfaces

In this paper we relate volumes of moduli spaces of super Riemann surfaces to integrals over the moduli space of stable Riemann surfaces $\overline{\cal M}_{g,n}$. This allows us to prove via algebraic geometry a recursion between the volumes of moduli spaces of super hyperbolic surfaces previously proven via super geometry techniques by Stanford and Witten. The recursion between the volumes of moduli spaces of super hyperbolic surfaces is proven to be equivalent to the fact that a generating function for the intersection numbers of a natural collection of cohomology classes $Θ_{g,n}$ with tautological classes on $\overline{\cal M}_{g,n}$ is a KdV tau function. This is analogous to Mirzakhani's proof of the Kontsevich-Witten theorem regarding a generating function for the intersection numbers of tautological classes on $\overline{\cal M}_{g,n}$ using volumes of moduli spaces of hyperbolic surfaces.

math.AG↗

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.

math.AG↗

arXiv2025

Volumes of moduli spaces of hyperbolic surfaces with cone points

In this paper we study volumes of moduli spaces of hyperbolic surfaces with geodesic, cusp and cone boundary components. We compute the volumes in some new cases, in particular when there exists a large cone angle. This allows us to give geometric meaning to Mirzakhani's polynomials under substitution of imaginary valued boundary lengths, corresponding to hyperbolic cone angles, and to study the behaviour of the volume under the $2π$ limit of a cone angle.

math.AG↗

arXiv2025

Combinatorics and large genus asymptotics of the Brézin--Gross--Witten numbers

In this paper, we study combinatorial and asymptotic properties of some interesting rational numbers called the Brézin--Gross--Witten (BGW) numbers, which can be represented as the intersection numbers of psi and Theta classes on the moduli space of stable algebraic curves. In particular, we discover and prove the uniform large genus leading asymptotics of certain normalized BGW numbers, and give a new proof of the polynomiality phenomenon for the large genus asymptotics. We also propose, with extensive numerical data, several new conjectures including monotonicity and integrality on the BGW numbers. Applications to the Painlevé II hierarchy and to the BGW-kappa numbers are given.

math-ph↗

arXiv2024

Super volumes and KdV tau functions

Weil-Petersson volumes of the moduli space of curves are deeply related to the Kontsevich-Witten KdV tau function. They possess a Virasoro symmetry which comes out of recursion relations between the volumes due to Mirzakhani. Similarly, the super Weil-Petersson volumes of the moduli space of super curves with Neveu-Schwarz punctures are related to the Brézin-Gross-Witten (BGW) tau function of the KdV hierarchy and satisfy a recursion due to Stanford and Witten, analogous to Mirzakhani's recursion. In this paper we prove that by also allowing Ramond punctures, the super Weil-Petersson volumes are related to the generalised BGW KdV tau function, which is a one parameter deformation of the BGW tau function. This allows us to prove that these new super volumes also satisfy the Stanford-Witten recursion.

math.AG↗

arXiv2024

Super Weil-Petersson measures on the moduli space of curves

The super Weil-Petersson metric defined over the moduli space of smooth super curves produces a natural measure over the moduli space of smooth curves. The construction of the measure uses the extra data of a spin structure on each smooth curve. When we allow marked points, the construction produces a collection of measures indexed by the behaviour of the spin structure at marked points -- Neveu-Schwarz or Ramond. In this paper we define these measures, and prove that they are finite. Each total measure gives the super volume of the moduli space of super curves with marked points. The Neveu-Schwarz volumes are polynomials that satisfy a recursion relation discovered by Stanford and Witten, analogous to Mirzakhani's recursion relations between Weil-Petersson volumes of moduli spaces of hyperbolic surfaces. We prove here that the Ramond boundary behaviour produces deformations of the Neveu-Schwarz volume polynomials, satisfying a variant of the Stanford-Witten recursion relations.

math.AG↗

arXiv2023

Weil-Petersson volumes, stability conditions and wall-crossing

In this paper we study Weil-Petersson volumes of the moduli spaces of conical hyperbolic surfaces. The moduli spaces are parametrised by their cone angles which naturally live inside Hassett's space of stability conditions on nodal curves. Such stability conditions produce weighted pointed stable curves which define compactifications of the moduli space of curves generalising the Deligne-Mumford compactification. The space of stability conditions decompose into chambers separated by walls. We assign to each chamber a polynomial corresponding to the Weil-Petersson volume of a moduli space of conical hyperbolic surfaces. The chambers are naturally partially ordered and the maximal chamber is assigned Mirzakhani's polynomial. We calculate wall-crossing polynomials, which relates the polynomial on any chamber to Mirzakhani's polynomial via wall-crossings, and we show how to apply this in particular cases. Since the polynomials are volumes, they have nice properties such as positivity, continuity across walls, and vanishing in certain limits.

math.AG↗

arXiv2021

Polynomial relations among kappa classes on the moduli space of curves

We construct an infinite collection of universal -- independent of $(g,n)$ -- polynomials in the Miller-Morita-Mumford classes $κ_m\in H^{2m}( \overline{\cal M}_{g,n},\bq)$, defined over the moduli space of genus $g$ stable curves with $n$ labeled points. We conjecture vanishing of these polynomials in a range depending on $g$ and $n$.

math.AG↗

arXiv2021

An intersection-theoretic proof of the Harer-Zagier formula

We provide an intersection-theoretic formula for the Euler characteristic of the moduli space of smooth curves. This formula reads purely in terms of Hodge integrals and, as a corollary, the standard calculus of tautological classes gives a new short proof of the Harer-Zagier formula. Our result is based on the Gauss-Bonnet formula, and on the observation that a certain parametrisation of the $Ω$-class - the Chern class of the universal $r$-th root of the twisted log canonical bundle - provides the Chern class of the log tangent bundle to the moduli space of smooth curves. Being $Ω$-classes by now employed in many enumerative problems, mostly recently found and at times surprisingly different from each other, we dedicate some work to produce an extensive list of their general properties: extending existing ones, finding new ones, and writing down some only known to the experts.

math.AG↗

arXiv2021

A new cohomology class on the moduli space of curves

We define a collection $Θ_{g,n}\in H^{4g-4+2n}(\overline{\cal M}_{g,n},\mathbb{Q})$ for $2g-2+n>0$ of cohomology classes that restrict naturally to boundary divisors. We prove that the intersection numbers $\int_{\overline{\cal M}_{g,n}}Θ_{g,n}\prod_{i=1}^nψ_i^{m_i}$ can be recursively calculated. We conjecture that a generating function for these intersection numbers is a tau function of the KdV hierarchy. This is analogous to the conjecture of Witten proven by Kontsevich that a generating function for the intersection numbers $\int_{\overline{\cal M}_{g,n}}\prod_{i=1}^nψ_i^{m_i}$ is a tau function of the KdV hierarchy.

math.AG↗

arXiv2020

Airy structures and deformations of curves in surfaces

An embedded curve in a symplectic surface $Σ\subset X$ defines a smooth deformation space $\mathcal{B}$ of nearby embedded curves. A key idea of Kontsevich and Soibelman arXiv:1701.09137 [math.AG], is to equip the symplectic surface $X$ with a foliation in order to study the deformation space $\mathcal{B}$. The foliation, together with a vector space $V_Σ$ of meromorphic differentials on $Σ$, endows an embedded curve $Σ$ with the structure of the initial data of topological recursion, which defines a collection of symmetric tensors on $V_Σ$. Kontsevich and Soibelman define an Airy structure on $V_Σ$ to be a formal quadratic Lagrangian $\mathcal{L}\subset T^*(V_Σ^*)$ which leads to an alternative construction of the tensors of topological recursion. In this paper we produce a formal series $θ$ on $\mathcal{B}$ of meromorphic differentials on $Σ$ which takes it values in $\mathcal{L}$, and use this to produce the Donagi-Markman cubic from a natural cubic tensor on $V_Σ$, giving a generalisation of a result of Baraglia and Huang, arXiv:1707.04975 [math.DG].

math.AG↗

arXiv2020

Gromov-Witten invariants of $\mathbb{P}^1$ coupled to a KdV tau function

We consider the pull-back of a natural sequence of cohomology classes $Θ_{g,n}\in H^{2(2g-2+n)}(\overline{\cal M}_{g,n})$ to the moduli space of stable maps ${\cal M}^g_n(\mathbb{P}^1,d)$. These classes are related to the Brézin-Gross-Witten tau function of the KdV hierarchy via $Z^{BGW}(\hbar,t_0,t_1,...)=\exp\sum\frac{\hbar^{2g-2}}{n!}\int_{\overline{\cal M}_{g,n}}Θ_{g,n}\cdot\prod_{j=1}^nψ_j^{k_j}\prod t_{k_j}$. Insertions of the pull-backs of the classes $Θ_{g,n}$ into the integrals defining Gromov-Witten invariants define new invariants which we show in the case of target $\mathbb{P}^1$ are given by a random matrix integral and satisfy the Toda equation.

math.AG↗

arXiv2019

JNR Monopoles

We review the theory of JNR, mass 1/2 hyperbolic monopoles in particular their spectral curves and rational maps. These are used to establish conditions for a spectral curve to be the spectral curve of a JNR monopole and to show that that rational map of a JNR monopole monopole arises by scattering using results of Atiyah. We show that for JNR monopoles the holomorphic sphere has a remarkably simple form and show that this can be used to give a formula for the energy density at infinity. In conclusion we illustrate some examples of the energy-density at infinity of JNR monopoles.

math.DG↗

arXiv2019

Loop equations for Gromov-Witten invariants of $\mathbb{P}^1$

We show that non-stationary Gromov-Witten invariants of $\mathbb{P}^1$ can be extracted from open periods of the Eynard-Orantin topological recursion correlators $ω_{g,n}$ whose Laurent series expansion at $\infty$ compute the stationary invariants. To do so, we overcome the technical difficulties to global loop equations for the spectral $x(z) = z + 1/z$ and $y(z) = \ln z$ from the local loop equations satisfied by the $ω_{g,n}$, and check these global loop equations are equivalent to the Virasoro constraints that are known to govern the full Gromov-Witten theory of $\mathbb{P}^1$.

math.AG↗

arXiv2018

Topological recursion with hard edges

We prove a Givental type decomposition for partition functions that arise out of topological recursion applied to spectral curves. Copies of the Konstevich-Witten KdV tau function arise out of regular spectral curves and copies of the Brezin-Gross-Witten KdV tau function arise out of irregular spectral curves. We present the example of this decomposition for the matrix model with two hard edges and spectral curve $(x^2-4)y^2=1$

math.AG↗

arXiv2016

Dubrovin's superpotential as a global spectral curve

We apply the spectral curve topological recursion to Dubrovin's universal Landau-Ginzburg superpotential associated to a semi-simple point of any conformal Frobenius manifold. We show that under some conditions the expansion of the correlation differentials reproduces the cohomological field theory associated with the same point of the initial Frobenius manifold.

math-ph↗