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

Mohamad Alameddine, Olivier Marchal

Abstract

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 its 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 relating the Jimbo-Miwa-Ueno differential as the $\hbar=0$ evaluation of the Hamiltonian differential in these Darboux coordinates that could provide insights on the geometric interpretation of the $\hbar$ formal parameter. As a byproduct we also obtain a rank $3$ Lax pair for the Painlevé IV equation.

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

Abstract

In this article, we present an explicit study of -deformed meromorphic connections in with an unramified irregular pole at infinity of order and its spectral dual corresponding to the 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 relating the Jimbo-Miwa-Ueno differential as the evaluation of the Hamiltonian differential in these Darboux coordinates that could provide insights on the geometric interpretation of the formal parameter. As a byproduct we also obtain a rank Lax pair for the Painlevé IV equation.
Paper Structure (39 sections, 21 theorems, 205 equations, 2 figures)

This paper contains 39 sections, 21 theorems, 205 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Meromorphic connections in $\mathfrak{gl}_3(\mathbb{C})$ with an irregular pole at infinity
  3. Representative normalized at infinity
  4. Isomonodromic deformations and Darboux coordinates
  5. Explicit expressions of the Lax matrices
  6. Monodromy-preserving deformations and solving the compatibility equation
  7. General monodromy-preserving deformations
  8. Compatibility equations and Hamiltonian evolutions
  9. Symplectic reduction of the tangent space
  10. Shifted Darboux coordinates and definition of trivial and non-trivial times
  11. Reduced Hamiltonian evolutions
  12. The Jimbo-Miwa-Ueno isomonodromic $\tau$-function
  13. Canonical choice of trivial times and reduced Lax matrices
  14. A Hermitian two-matrix model for the $\mathfrak{gl}_3$ side and relation with the JMU tau-function
  15. Classical spectral curve and $\mathfrak{gl}_3$ duality gauge
  16. ...and 24 more sections

Key Result

Proposition 2.1

The gauge matrix $G(\lambda)$ such that $\Psi(\lambda)=G(\lambda) \tilde{\Psi}(\lambda)$ is given by where the last line of the gauge matrix is given by The Lax matrix normalized at infinity is given by: where $\tilde{L}^{[\infty,0]}$ has the following form with The Lax matrix $L(\lambda)$ in the oper gauge is characterized by its non-trivial last line: where the polynomials $P_1(\lambda)$,

Figures (2)

  • Figure 1: Summary of the symplectic structures associated with the meromorphic connections on both sides and their relations by duality.
  • Figure 2: Newton polygons on both sides. The left figure represents the Newton polygons of $\mathcal{S}_d$ (light blue) or $\mathcal{S}$ (light and dark blue). The right figure represents the Newton polygon of $\mathcal{S}_{\text{P4}}$ under the constraint $s_{X_1^{(1)},0}s_{X_1^{(2)},0}=0$.

Theorems & Definitions (69)

  • Definition 2.1: Space of connections
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.1: Lax matrix and gauge matrix
  • proof
  • Remark 2.7
  • ...and 59 more