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

Olivier Marchal, Mohamad Alameddine

TL;DR

This work develops a complete Hamiltonian framework for isomonodromic deformations of twisted $\mathfrak{gl}_2(\mathbb{C})$ connections with ramified irregular poles at infinity, tying the Painlevé $1$ hierarchy to explicit Lax pairs and Hamiltonians expressed in irregular times and Darboux coordinates. A central advance is the reduction of the deformation space from $2g+4$ irregular times to a distinguished $g$-dimensional set of isomonodromic times, together with a symplectic change to symmetric Darboux coordinates that yields polynomial Hamiltonians and Lax matrices. The theory connects to the classical spectral curve and topological recursion, and is concretely demonstrated in the Airy case and the first three Painlevé $1$ hierarchy members, including canonical Lax pairs and explicit Hamiltonians. This provides a scalable, algebraic toolkit for explicit isomonodromic dynamics in twisted settings and opens pathways to generalizations to higher rank and links with quantum and topological-REC structures. All results are presented with explicit formulae in terms of $\hbar$, irregular times $t_{\infty,k}$, and Darboux coordinates, making them amenable to computational and SEO indexing with precise mathematical notation.

Abstract

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.

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

TL;DR

This work develops a complete Hamiltonian framework for isomonodromic deformations of twisted connections with ramified irregular poles at infinity, tying the Painlevé hierarchy to explicit Lax pairs and Hamiltonians expressed in irregular times and Darboux coordinates. A central advance is the reduction of the deformation space from irregular times to a distinguished -dimensional set of isomonodromic times, together with a symplectic change to symmetric Darboux coordinates that yields polynomial Hamiltonians and Lax matrices. The theory connects to the classical spectral curve and topological recursion, and is concretely demonstrated in the Airy case and the first three Painlevé hierarchy members, including canonical Lax pairs and explicit Hamiltonians. This provides a scalable, algebraic toolkit for explicit isomonodromic dynamics in twisted settings and opens pathways to generalizations to higher rank and links with quantum and topological-REC structures. All results are presented with explicit formulae in terms of , irregular times , and Darboux coordinates, making them amenable to computational and SEO indexing with precise mathematical notation.

Abstract

In this paper, we build the Hamiltonian system and the corresponding Lax pairs associated to a twisted connection in admitting an irregular and ramified pole at infinity of arbitrary degree, hence corresponding to the Painlevé hierarchy. We provide explicit formulas for these Lax pairs and Hamiltonians in terms of the irregular times and standard Darboux coordinates associated to the twisted connection. Furthermore, we obtain a map that reduces the space of irregular times to only 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 , the Painlevé case and the next two elements of the Painlevé hierarchy.
Paper Structure (51 sections, 33 theorems, 295 equations)

This paper contains 51 sections, 33 theorems, 295 equations.

Table of Contents

  1. Introduction and summary of the results
  2. Twisted meromorphic connections at infinity
  3. Twisted meromorphic connections and irregular times
  4. Choice of representative normalized at infinity
  5. Darboux coordinates
  6. Scalar differential equation and companion gauge
  7. Introduction of a scaling parameter $\hbar$
  8. Explicit expressions of the gauge transformation
  9. Wronksians and asymptotics of the wave functions
  10. Explicit expression for the Lax matrix $L$
  11. Classical spectral curve and connection with topological recursion
  12. General isomonodromic deformations and auxiliary matrices
  13. Definition of general isomonodromic deformations
  14. General form of the auxiliary matrix $A_{\boldsymbol{\alpha}}(\lambda,\hbar)$
  15. General Hamiltonian evolutions
  16. ...and 36 more sections

Key Result

Proposition 2.1

Let $z\overset{\text{def}}{:=} \lambda^{\frac{1}{2}}$. For any given $\hat{L}(\lambda)$ in an orbit of $\hat{F}_{\infty,r_\infty}$, there exists a local gauge matrix $G_\infty(z)$ around $\infty$ such that and The complex numbers $\left(t_{\infty,k}\right)_{1\leq k\leq 2r_\infty-2}$ define the "irregular times" at infinity that we shall denote $\mathbf{t}=\{(t_{\infty,k})_{1\leq k\leq 2r_\infty-

Theorems & Definitions (74)

  • Definition 2.1: Space of meromorphic connections with a pole at infinity
  • Definition 2.2: Set of twisted meromorphic connections at infinity
  • Proposition 2.1
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.3
  • proof
  • ...and 64 more