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Alexander Hock

Publications and source records attributed to Alexander Hock.

At least 19 recordsLinked to original sources
arXiv2026

Universal Correlators on Exponentially Ramified Spectral Curves

We investigate generalized topological recursion on compact spectral curves admitting exponential, and more generally essential, singularities as ramification points. Exploiting the global formulation of generalized topological recursion, we establish a contour deformation of the recursive residue formula that replaces contributions from these essential singularities by residues at meromorphic points. This provides a natural recursive framework for exponentially ramified spectral curves while remaining entirely within the generalized topological recursion formalism. Our formalism can also be viewed as a limiting case of the Bouchard-Eynard higher-order topological recursion, obtained when the order of a ramification point tends to infinity in a convergent manner, as occurs, for example, for exponential singularities while $dy$ remains regular and non-vanishing. We further illustrate the resulting formalism through several examples, including transcendental functions and the $x$-$y$ dual of the Mirzakhani curve.

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arXiv2026

Geometry of Logarithmic Topological Recursion: Dilaton Equations, Free Energies and Variational Formulas

One of the most important applications of topological recursion concerns spectral curves for which the functions $(x,y)$ defining the spectral curve are allowed to have logarithmic singularities. This occurs for instance for Seiberg-Witten curves and mirror curves computing Gromov--Witten invariants of toric Calabi--Yau threefolds. A recently introduced extension of topological recursion, the so-called logarithmic topological recursion, exhibits the correct behavior under certain limits of those spectral curves. In this article, we derive the dilaton equations in the setting of logarithmic topological recursion, as well as variational formulas, and provide a definition of the free energies in situations where standard topological recursion was known to fail. We present examples in which the new definition of the free energies \textit{directly} (without any computation) reproduces the full perturbative part of the Nekrasov--Shatashvili partition function of 4d $\mathcal{N}=2$ pure supersymmetric gauge theory, as well as the all-genus free energies of mirror curves of strip geometries, including in particular the topological vertex and the resolved conifold.

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arXiv2025

Quantum Curve for strip geometries, Topological Recursion and open GW/DT invariants

Open topological string partition function gives rise to open Gromov-Witten invariants, open Donaldson-Thomas invariants and 3D-5D BPS indices. Utilizing the remodelling conjecture which connects topological recursion and topological string theory, in this paper we study open topological string theory for the subclass of toric Calabi-Yau threefold known as strip geometries. For this purpose, certain new developments in the theory of topological recursion are applied as its extension to Logarithmic Topological Recursion (Log-TR) and the universal $x$--$y$ duality. Through this we derive the open topological string partition function and also the associated quantum curve. We also explain how this is related to the open Donaldson-Thomas partition function associated with certain symmetric quivers, exponential networks and $q$-Barnes type integrals. In the process, we also connect how 3D-5D wall crossing affects these partition functions as one varies $x$, in examples.

math-ph↗
arXiv2025

GW/DT invariants and 5D BPS indices for strips from topological recursion

Topological string theory partition function gives rise to Gromov-Witten invariants, Donaldson-Thomas invariants and 5D BPS indices. Using the remodeling conjecture, which connects Topological Recursion with topological string theory for toric Calabi-Yau threefolds, we study a more direct connection for the subclass of strip geometries. In doing so, new developments in the theory of topological recursion are applied as its extension to Logarithmic Topological Recursion (Log-TR) and the universal $x$-$y$ duality. Through these techniques, our main result in this paper is a direct derivation of all free energies from topological recursion for general strip geometries. In analyzing the expression of free energy, we shed some light on the meaning and the influence of the $x$-$y$ duality in topological string theory and its interconnection to GW and DT invariants as well as the 5D BPS index.

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arXiv2025

Quantum Curves in the Context of Symplectic Duality

We discuss how to use the recent progress in understanding of the $x$-$y$ duality and symplectic duality in the theory of topological recursion and its generalizations in order to efficiently compute the quantum spectral curve operators for the wave functions with arbitrary base points. The paper also contains an overview of recent generalizations of the setup of topological recursion prompted by the progress in understanding the $x$-$y$ duality.

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arXiv2025

Symplectic (Non-)Invariance of the Free Energy in Topological Recursion

Let $F_g$ be the free energy derived from Topological Recursion for a given spectral curve on a compact Riemann surface, and let $F_g^\vee$ be its $x$-$y$ dual, that is, the free energy derived from the same spectral curve with the roles of $x$ and $y$ interchanged. $F_g$ is sometimes called a symplectic invariant due to its invariance under certain symplectomorphisms of the formal symplectic form $dx\wedge dy$. However, the free energy is not generally invariant under the swap of $x$ and $y$; thus, the difference $F_g - F_g^\vee$ is nonzero. We derive a new formula for this difference for all $g\geq 2$ in terms of a residue calculation at the singularities of $x$ and $y$, including cases where $x$ and $y$ have logarithmic singularities. For the derivation, we apply recent developments from $x$-$y$ duality within the theory of (Logarithmic) Topological Recursion. The derived formulas are particularly useful for spectral curves with a trivial $x$-$y$ dual side, meaning those with vanishing $F_g^\vee$. In such cases, one obtains an explicit result for $F_{g\geq 2}$ itself. We apply this to several classes of spectral curves and prove, for instance, a recent conjecture by Borot et al. that the free energies $F_g$ computed by Topological Recursion for the "Gaiotto curve" coincide with the perturbative part (in the $\Omega$-background) of the Nekrasov partition function of $\mathcal{N}=2$ pure supersymmetric gauge theory. Similar computations also provide $F_g$ for the CDO curve related to Hurwitz numbers, or the negative $r$-spin curve related to $\Theta$-class intersection numbers on $\overline{\mathcal{M}}_{g,n}$.

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arXiv2024

Bialgebras, and Lie monoid actions in Morse and Floer theory, I

We introduce a new family of oriented manifolds with boundaries called the forest biassociahedra and forest bimultiplihedra, generalizing the standard biassociahedra. They are defined as moduli spaces of ascending-descending biforests and are expected to act as parameter spaces for operations defined on Morse and Floer chains in the context of compact Lie group actions. We study the structure of their boundary, and derive some algebraic notions of ``$f$-bialgebras'', as well as related notions of bimodules, morphisms and categories. This allows us to state some conjectures describing compact Lie group actions on Morse and Floer chains, and on Fukaya categories.

math.SG↗
arXiv2024

Combinatorial Dyson-Schwinger Equations of Quartic Matrix Field Theory

Matrix field theory is a combinatorially non-local field theory which has recently been found to be a non-trivial but solvable QFT example. To generalize such non-perturbative structures to other models, a more combinatorial understanding of Dyson-Schwinger equations and their solutions is of high interest. To this end we consider combinatorial Dyson-Schwinger equations manifestly relying on the Hopf-algebraic structure of perturbative renormalization. We find that these equations are fully compatible with renormalization, relying only on the superficially divergent diagrams which are planar ribbon graphs, i.e. decompleted dual combinatorial maps. Still, they are of a similar kind as in realistic models of local QFT, featuring in particular an infinite number of primitive diagrams as well as graph-dependent combinatorial factors.

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arXiv2023

$x-y$ duality in Topological Recursion for exponential variables via Quantum Dilogarithm

For a given spectral curve, the theory of topological recursion generates two different families $\omega_{g,n}$ and $\omega_{g,n}^\vee$ of multi-differentials, which are for algebraic spectral curves related via the universal $x-y$ duality formula. We propose a formalism to extend the validity of the $x-y$ duality formula of topological recursion from algebraic curves to spectral curves with exponential variables of the form $e^x=F(e^y)$ or $e^x=F(y)e^{a y}$ with $F$ rational and $a$ some complex number, which was in principle already observed in \cite{Dunin-Barkowski:2017zsd,Bychkov:2020yzy}. From topological recursion perspective the family $\omega_{g,n}^\vee$ would be trivial for these curves. However, we propose changing the $n=1$ sector of $\omega_{g,n}^\vee$ via a version of the Faddeev's quantum dilogarithm which will lead to the correct two families $\omega_{g,n}$ and $\omega_{g,n}^\vee$ related by the same $x-y$ duality formula as for algebraic curves. As a consequence, the $x-y$ symplectic transformation formula extends further to important examples governed by topological recursion including, for instance, the topological vertex curve which computes Gromov-Witten invariants of $\mathbb{C}^3$, equivalently triple Hodge integrals on the moduli space of complex curves, orbifold Hurwitz numbers, or stationary Gromov-Witten invariants of $\mathbb{P}^1$. The proposed formalism is related to the issue topological recursion encounters for specific choices of framings for the topological vertex curve.

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arXiv2023

Genus Permutations and Genus Partitions

For a given permutation or set partition there is a natural way to assign a genus. Counting all permutations or partitions of a fixed genus according to cycle lengths or block sizes, respectively, is the main content of this article. After a variable transformation, the generating series are rational functions with poles located at the ramification points in the new variable. The generating series for any genus is given explicitly for permutations and up to genus 2 for set partitions. Extending the topological structure not just by the genus but also by adding more boundaries, we derive the generating series of non-crossing partitions on the cylinder from known results of non-crossing permutations on the cylinder. Most, but not all, outcomes of this article are special cases of already known results, however they are not represented in this way in the literature, which however seems to be the canonical way. To make the article as accessible as possible, we avoid going into details into the explicit connections to Topological Recursion and Free Probability Theory, where the original motivation came from.

math.CO↗
arXiv2023

Laplace transform of the $x-y$ symplectic transformation formula in Topological Recursion

The functional relation coming from the $x-y$ symplectic transformation of Topological Recursion has a lot of applications, for instance it is the higher order moment-cumulant relation in free probability or can be used to compute intersection numbers on the moduli space of complex curves. We derive the Laplace transform of this functional relation, which has a very nice and compact form as a formal power series in $\hbar$. We apply the Laplace transformed formula to the Airy curve and the Lambert curve.

math-ph↗
arXiv2023

Blobbed topological recursion from extended loop equations

We consider the $N\times N$ Hermitian matrix model with measure $d\mu_{E,\lambda}(M)=\frac{1}{Z} \exp(-\frac{\lambda N}{4} \mathrm{tr}(M^4)) d\mu_{E,0}(M)$, where $d\mu_{E,0}$ is the Gaussian measure with covariance $\langle M_{kl}M_{mn}\rangle=\frac{\delta_{kn}\delta_{lm}}{N(E_k+E_l)}$ for given $E_1,...,E_N>0$. It was previously understood that this setting gives rise to two ramified coverings $x,y$ of the Riemann sphere strongly tied by $y(z)=-x(-z)$ and a family $\omega^{(g)}_{n}$ of meromorphic differentials conjectured to obey blobbed topological recursion due to Borot and Shadrin. We develop a new approach to this problem via a system of six meromorphic functions which satisfy extended loop equations. Two of these functions are symmetric in the preimages of $x$ and can be determined from their consistency relations. An expansion at $\infty$ gives global linear and quadratic loop equations for the $\omega^{(g)}_{n}$. These global equations provide the $\omega^{(g)}_{n}$ not only in the vicinity of the ramification points of $x$ but also in the vicinity of all other poles located at opposite diagonals $z_i+z_j=0$ and at $z_i=0$. We deduce a recursion kernel representation valid at least for $g\leq 1$.

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arXiv2022

A simple formula for the $x$-$y$ symplectic transformation in topological recursion

Let $W_{g,n}$ be the correlators computed by Topological Recursion for some given spectral curve $(x,y)$ and $W^\vee_{g,n}$ for $(y,x)$, where the role of $x,y$ is inverted. These two sets of correlators $W_{g,n}$ and $W^\vee_{g,n}$ are related by the $x$-$y$ symplectic transformation. Bychkov, Dunin-Barkowski, Kazarian and Shadrin computed a functional relation between two slightly different sets of correlators. Together with Alexandrov, they proved that their functional relation is indeed the $x$-$y$ symplectic transformation in Topological Recursion. This article provides a fairly simple formula directly between $W_{g,n}$ and $W^\vee_{g,n}$ which holds by their theorem for meromorphic $x$ and $y$ with simple and distinct ramification points. Due to the recent connection between free probability and fully simple vs ordinary maps, we conclude a simplified moment-cumulant relation for moments and higher order free cumulants.

math-ph↗
arXiv2022

Complete solution of the LSZ Model via Topological Recursion

We prove that the Langmann-Szabo-Zarembo (LSZ) model with quartic potential, a toy model for a quantum field theory on noncommutative spaces grasped as a complex matrix model, obeys topological recursion of Chekhov, Eynard and Orantin. By introducing two families of correlation functions, one corresponding to the meromorphic differentials $\omega_{g,n}$ of topological recursion, we obtain Dyson-Schwinger equations that eventually lead to the abstract loop equations being, together with their pole structure, the necessary condition for topological recursion. This strategy to show the exact solvability of the LSZ model establishes another approach towards the exceptional property of integrability in some quantum field theories. We compare differences in the loop equations for the LSZ model (with complex fields) and the Grosse-Wulkenhaar model (with hermitian fieldss) and their consequences for the resulting particular type of topological recursion that governs the models.

math-ph↗
arXiv2022

An irregular spectral curve for the generation of bipartite maps in topological recursion

We derive an efficient way to obtain generating functions of bipartite maps of arbitrary genus and boundary length using a spectral curve as initial data for the framework of topological recursion. Based on an earlier result of Chapuy and Fang counting these maps and having a structural proximity to topological recursion, we deduce the corresponding spectral curve which has a strong relation to the spectral curve giving rise to generating functions of ordinary maps. In contrast to ordinary maps, the spectral curve is an irregular one in the sense of Do and Norbury. It generalises the irregular curve for the enumeration of Grothendieck's dessins d'enfant.

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arXiv2022

On the $x$-$y$ Symmetry of Correlators in Topological Recursion via Loop Insertion Operator

Topological Recursion generates a family of symmetric differential forms (correlators) from some initial data $(\Sigma,x,y,B)$. We give a functional relation between the correlators of genus $g=0$ generated by the initial data $(\Sigma,x,y,B)$ and by the initial data $(\Sigma,y,x,B)$, where $x$ and $y$ are interchanged. The functional relation is derived with the loop insertion operator by computing a functional relation for some intermediate correlators. Additionally, we show that our result is equivalent to the recent result of \cite{Borot:2021thu} in case of $g=0$. Consequently, we are providing a simplified functional relation between generating series of higher order free cumulants and moments in higher order free probability.

math-ph↗
arXiv2021

Genus one free energy contribution to the quartic Kontsevich model

We prove a formula for the genus one free energy $\mathcal{F}^{(1)}$ of the quartic Kontsevich model for arbitrary ramification by working out a boundary creation operator for blobbed topological recursion. We thus investigate the differences in $\mathcal{F}^{(1)}$ compared with its generic representation for ordinary topological recursion. In particular, we clarify the role of the Bergman $\tau$-function in blobbed topological recursion. As a by-product, we show that considering the holomorphic additions contributing to $\omega_{g,1}$ or not gives a distinction between the enumeration of bipartite and non-bipartite quadrangulations of a genus-$g$ surface.

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