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Simply-laced isomonodromy systems

Philip Boalch

TL;DR

This work develops a unified, graph-theoretic framework for isomonodromic deformations that extends classical Painlevé equations to higher-rank and higher-order systems. By introducing supernova graphs and a Weyl-algebra/module viewpoint, the author constructs a family of isomonodromy systems with Weyl-group symmetries coming from attached Kac–Moody root systems, and proves invariance under ${\rm SL}_2(\mathbb{C})$-actions and related gauge symmetries. The approach yields concrete nonlinear equations that generalize JMMS and Schlesinger equations, with several explicit realizations (JMMS, Schlesinger, dual Schlesinger) and a full reduction to meromorphic connections on trivial bundles. The framework predicts a rich collection of higher Painlevé systems ($\text{hP}^{n}_{VI}$, $\text{hP}^{n}_{V}$, $\text{hP}^{n}_{IV}$, $\text{hP}^{n}_{III}$, $\text{hP}^{n}_{II}$) associated to complete $k$-partite graphs, and posits a conjectural link between Hitchin-Hilbert schemes and these deformations. Overall, the paper provides both a rigorous geometric–algebraic foundation and a practical catalogue of higher Painlevé-type isomonodromic systems with deep connections to nonabelian Hodge theory and quiver varieties.

Abstract

A new class of isomonodromy equations will be introduced and shown to admit Kac-Moody Weyl group symmetries. This puts into a general context some results of Okamoto on the 4th, 5th and 6th Painleve equations, and shows where such Kac-Moody root systems occur "in nature". A key point is that one may go beyond the class of affine Kac-Moody root systems. As examples, by considering certain hyperbolic Kac-Moody Dynkin diagrams, we find there is a sequence of higher order Painleve systems lying over each of the classical Painleve equations. This leads to a conjecture about the Hilbert scheme of points on some Hitchin systems.

Simply-laced isomonodromy systems

TL;DR

This work develops a unified, graph-theoretic framework for isomonodromic deformations that extends classical Painlevé equations to higher-rank and higher-order systems. By introducing supernova graphs and a Weyl-algebra/module viewpoint, the author constructs a family of isomonodromy systems with Weyl-group symmetries coming from attached Kac–Moody root systems, and proves invariance under -actions and related gauge symmetries. The approach yields concrete nonlinear equations that generalize JMMS and Schlesinger equations, with several explicit realizations (JMMS, Schlesinger, dual Schlesinger) and a full reduction to meromorphic connections on trivial bundles. The framework predicts a rich collection of higher Painlevé systems (, , , , ) associated to complete -partite graphs, and posits a conjectural link between Hitchin-Hilbert schemes and these deformations. Overall, the paper provides both a rigorous geometric–algebraic foundation and a practical catalogue of higher Painlevé-type isomonodromic systems with deep connections to nonabelian Hodge theory and quiver varieties.

Abstract

A new class of isomonodromy equations will be introduced and shown to admit Kac-Moody Weyl group symmetries. This puts into a general context some results of Okamoto on the 4th, 5th and 6th Painleve equations, and shows where such Kac-Moody root systems occur "in nature". A key point is that one may go beyond the class of affine Kac-Moody root systems. As examples, by considering certain hyperbolic Kac-Moody Dynkin diagrams, we find there is a sequence of higher order Painleve systems lying over each of the classical Painleve equations. This leads to a conjecture about the Hilbert scheme of points on some Hitchin systems.

Paper Structure

This paper contains 44 sections, 46 theorems, 220 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Further remarks and viewpoints
  3. Summary of main result
  4. Prototype: The lifted JMMS equations
  5. Generalisation
  6. Modules over the Weyl algebra
  7. Symplectic vector spaces
  8. Invariance of spectral invariants
  9. Time-dependent Hamiltonians
  10. The space of times
  11. Hamiltonians
  12. Further properties of the Hamiltonians
  13. Isomonodromy
  14. Full connections
  15. Nonlinear differential equations
  16. ...and 29 more sections

Key Result

Theorem 1.1

$\bullet$ There is an isomonodromy system where the space of times is $\mathbb{B}\cong \prod_{j\in J} (\mathbb{C}^{\lvert I_j\rvert}\setminus {\mathop{\rm diagonals}}).$ It controls isomonodromic deformations of certain linear differential systems on bundles of rank where $I_\infty\subset I$ is the part with $a_j=\infty$. $\bullet$ If $i\in \widehat{I}$ is a node of $\widehat{\mathcal{G}}$ and $

Figures (12)

  • Figure 1: Affine Dynkin diagrams for Painlevé equations IV, V and VI.
  • Figure 2: Dynkin diagrams for simply-laced higher Painlevé systems.
  • Figure 3: Complete $k$-partite graphs from partitions of $N\le 6$
  • Figure 4: Splaying both nodes.
  • Figure 5: How $A_3^{(1)}$ appears in the graphical approach to the JMMS system.
  • ...and 7 more figures

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2: Harnad duality Harn94
  • Lemma 2.1
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 4.1
  • Definition 5.1
  • Lemma 5.2
  • Lemma 5.3
  • Theorem 5.4
  • ...and 47 more