arXiv subjects

Mohamad Alameddine

Publications and source records attributed to Mohamad Alameddine.

11 recordsLinked to original sources

arXiv2026

Hamiltonian representation of isomonodromic deformations of general rational connections on $\mathfrak{gl}_2(\mathbb{C})$

In this paper, we construct the Hamiltonian systems attached to generic $\mathfrak{gl}_2(\mathbb{C})$ meromorphic connections with an arbitrary number of unramified poles of arbitrary orders, under the assumption that every leading polar coefficient is regular semisimple. In particular, we propose the Lax pairs and Hamiltonian evolutions expressed in terms of irregular times and monodromies associated to the poles as well as $g$ pairs of Darboux coordinates defined as the apparent singularities arising in the oper gauge. Moreover, we also provide a reduction of the isomonodromic deformations to a subset of $g$ non-trivial isomonodromic deformations. This reduction is equivalent to a map reducing the set of irregular times to only $g$ non-trivial isomonodromic times. We apply our construction to all genus-one cases covered by these hypotheses and recover the standard Painlevé equations $2$--$6$. We finally make the connection with the topological recursion and the quantization of classical spectral curves from this perspective.

math-ph↗

arXiv2026

Hamiltonian representation of isomonodromic deformations of twisted rational connections: The Painlevé $1$ hierarchy

In this paper, we build the Hamiltonian system and the corresponding Lax pairs associated to a twisted connection in $\mathfrak{gl}_2(\mathbb{C})$ admitting an irregular and ramified pole at infinity of arbitrary degree, hence corresponding to the Painlevé $1$ hierarchy. We provide explicit formulas for these Lax pairs and Hamiltonians in terms of the irregular times and standard $2g$ Darboux coordinates associated to the twisted connection. Furthermore, we obtain a map that reduces the space of irregular times to only $g$ non-trivial isomonodromic deformations. In addition, we perform a symplectic change of Darboux coordinates to obtain a set of symmetric Darboux coordinates in which Hamiltonians and Lax pairs are polynomial. Finally, we apply our general theory to the first cases of the hierarchy: the Airy case $(g=0)$, the Painlevé $1$ case $(g=1)$ and the next two elements of the Painlevé $1$ hierarchy.

math-ph↗

arXiv2026

The Painlevé I hierarchy: Correspondence between the isomonodromic approach and the minimal models of the KP hierarchy

Two approaches to the Painlevé I hierarchy are discussed: the isomonodromic construction based on meromorphic connections, and the minimal models construction based on a reduction of the KP hierarchy. An explicit correspondence between both formalisms is established, identifying these setups explicitly. In particular, this yields new expressions for the Lax matrices and Hamiltonians.

math-ph↗

arXiv2026

Explicit Hamiltonian representations of meromorphic connections and duality from different perspectives: a case study

In this article, we present an explicit study of $\hbar$-deformed meromorphic connections in $\mathfrak{gl}_3(\mathbb{C})$ with an unramified irregular pole at infinity of order $r_\infty=3$ and their spectral dual corresponding to the $\mathfrak{gl}_2(\mathbb{C})$ Painlevé IV Lax pair. Using the apparent singularities and their dual partners on the spectral curves as Darboux coordinates, we obtain the Hamiltonian evolutions, the reduction of these evolutions to a single non-trivial direction, the Jimbo-Miwa-Ueno tau-functions, the fundamental symplectic two-forms and the associated Hermitian matrix models on both sides. We then prove that the spectral duality connecting both sides extends to all these aspects, providing an explicit illustration of the generalized Harnad duality. We finally propose a conjecture identifying the Jimbo-Miwa-Ueno differential with the $\hbar=0$ evaluation of the Hamiltonian differential in these Darboux coordinates that could provide insights into the geometric interpretation of the $\hbar$ formal parameter. As a byproduct we also obtain a rank $3$ Lax pair for the Painlevé IV equation.

math-ph↗

arXiv2026

Explicit Hamiltonian structure of the Flaschka-Newell Painlevé II hierarchy via symmetry reduction of the Painlevé IV hierarchy

We study the Hamiltonian structure of the Flaschka-Newell Painlevé II hierarchy via symmetry reduction of the associated space of meromorphic connections. Building on the realization of this hierarchy as a reduction of the Painlevé IV hierarchy via a $\mathbb{Z}_2$-symmetry, we construct a set of Darboux coordinates adapted to the involution. After a suitable change of trivialization, the symmetry acts diagonally in these coordinates, allowing the fixed-point locus to be explicitly described as a symplectic submanifold. This enables us to derive the reduced Hamiltonians after symmetry, thereby obtaining explicit expressions for the Hamiltonians and the Lax matrices of the Flaschka-Newell Painlevé II hierarchy. This strategy also illustrates how symmetry-adapted canonical Darboux coordinates enable explicit reductions of isomonodromic systems at the level of their underlying symplectic geometry.

math-ph↗

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.

math-ph↗

arXiv2026

A symmetry reduction of the Painlevé IV hierarchy to the Flaschka-Newell Painlevé II hierarchy

We study the isomonodromic deformation problem associated with rank-two meromorphic connections on the Riemann sphere having one regular singularity and one irregular singularity of even order at infinity, corresponding to the even Painlevé IV hierarchy. We show that the symmetry $Ψ(-λ)= σ_1 Ψ(λ) σ_1$ defines an invariant submanifold whose induced isomonodromic dynamics coincides with the Flaschka-Newell Painlevé II hierarchy. Under this identification, the corresponding Lax matrices, Darboux coordinates and Hamiltonian structures can be matched explicitly. In particular, the Hamiltonians of the first members of the Flaschka-Newell hierarchy are recovered from the even Painlevé IV hierarchy. This provides a geometric interpretation of the Flaschka-Newell hierarchy as a symmetry reduction of an isomonodromic deformation problem, complementing its classical description as a similarity reduction of the modified Korteweg-de Vries hierarchy.

math-ph↗

arXiv2025

A Twisted sl_2(C) Isomonodromic-Isospectral Correspondence

The aim of this article is to generalize the isomonodromic-isospectral correspondence for meromorphic connections of rank $2$ over $\mathbb{P}^1$ to the twisted case. More specifically, the construction of the isospectral approach is provided for the Painlevé I hierarchy, then two maps are constructed, one linking the sets of isomonodromic and isospectral Hamiltonians, another linking the set of apparent singularities to a set of isospectral coordinates.

math-ph↗

arXiv2025

Resurgence of Tritronquées Solutions of the Deformed Painlevé I Equation

We prove that the formal $\hbar$-power series solution of the deformed Painlevé I equation is resurgent, which means it is generically Borel summable and its Borel transform admits endless analytic continuation. In particular, we find that the Borel transform defines a global multivalued holomorphic function on a singular algebraic surface isomorphic to the Fermat quintic surface $x^5 + y^5 + z^5 = 0$ modulo an involution. This surface is an algebraic fibration over the complex plane of the differential equation with generic fibre a smooth quintic curve. Each fibre is equipped with a fivefold covering map over another complex plane (the Borel plane) with ten ramification points (the Borel singularities) spread equally over two branch points giving two opposite Stokes rays.

math.DG↗

arXiv2024

On the geometry of isomonodromic deformations on the torus and the elliptic Calogero-Moser system

We consider isomonodromic deformations of connections with a simple pole on the torus, motivated by the elliptic version of the sixth Painlevé equation. We establish an extended symmetry, complementing known results. The Calogero-Moser system in its elliptic version is shown to fit nicely in the geometric framework, the extended symplectic two-form is introduced and shown to be closed.

math-ph↗

arXiv2024

Isomonodromic and isospectral deformations of meromorphic connections: the $\mathfrak{sl}_2(\mathbb{C})$ case

We consider non-twisted meromorphic connections in $\mathfrak{sl}_2(\mathbb{C})$ and the associated symplectic Hamiltonian structure. In particular, we provide explicit expressions of the Lax pair in the geometric gauge supplementing previous results where explicit formulas have been obtained in the oper gauge. Expressing the geometric Lax matrices requires the introduction of specific Darboux coordinates for which we provide the explicit Hamiltonian evolutions. These expressions allow to build bridges between the isomonodromic deformations and the isospectral ones. More specifically, we propose an explicit change of Darboux coordinates to obtain isospectral coordinates for which Hamiltonians match the spectral invariants. This result solves the issue left opened in \cite{BertolaHarnadHurtubise2022} in the case of $\mathfrak{sl}_2(\mathbb{C})$.

math-ph↗