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Isoperiodic deformations of Toda curves and chains, the difference Korteweg - de Vries equation, and $SU(N)$ Seiberg-Witten theories

Vladimir Dragović, Vasilisa Shramchenko

TL;DR

The paper develops a comprehensive algebro-geometric framework for isoperiodic deformations of Toda curves and their higher-order generalizations. It constructs Toda families from Pell equation data, derives Rauch-based variational formulas, and obtains explicit isoperiodicity equations that govern the motion of dependent branch points, including generalized Toda families with $\Omega_\alpha$. The work connects these deformations to isospectral Jacobi problems, KdV difference equations, and SU($N$) Seiberg–Witten theories, showing how singular regimes and massless particles are preserved under isoperiodic flows. It also links equilibrium measures on interval unions to isoperiodic deformations and provides a concrete triangular solution to a constrained Schlesinger system, highlighting deep interactions between integrable systems, spectral theory, and gauge theory with explicit polynomial/Pell-structure underpinnings.

Abstract

We introduce the dynamics of Toda curves of order $N$ and derive differential equations governing this dynamics. We apply the obtained results to describe isoperiodic deformations of $N$-periodic Toda chains and periodic difference Korteweg-de Vries equation. We describe deformations of the essential spectra of $N$-periodic two-sided Jacobi matrices. We also study singular regimes of $SU(N)$ Seiberg-Witten theory and describe their deformations preserving the number of singularities where new massless particles may occur. We introduce and describe isoequilibrium deformations of arbitrary collections of $d$ real disjoint closed intervals. We conclude by providing explicit triangular solutions to constrained Schlesinger systems.

Isoperiodic deformations of Toda curves and chains, the difference Korteweg - de Vries equation, and $SU(N)$ Seiberg-Witten theories

TL;DR

The paper develops a comprehensive algebro-geometric framework for isoperiodic deformations of Toda curves and their higher-order generalizations. It constructs Toda families from Pell equation data, derives Rauch-based variational formulas, and obtains explicit isoperiodicity equations that govern the motion of dependent branch points, including generalized Toda families with . The work connects these deformations to isospectral Jacobi problems, KdV difference equations, and SU() Seiberg–Witten theories, showing how singular regimes and massless particles are preserved under isoperiodic flows. It also links equilibrium measures on interval unions to isoperiodic deformations and provides a concrete triangular solution to a constrained Schlesinger system, highlighting deep interactions between integrable systems, spectral theory, and gauge theory with explicit polynomial/Pell-structure underpinnings.

Abstract

We introduce the dynamics of Toda curves of order and derive differential equations governing this dynamics. We apply the obtained results to describe isoperiodic deformations of -periodic Toda chains and periodic difference Korteweg-de Vries equation. We describe deformations of the essential spectra of -periodic two-sided Jacobi matrices. We also study singular regimes of Seiberg-Witten theory and describe their deformations preserving the number of singularities where new massless particles may occur. We introduce and describe isoequilibrium deformations of arbitrary collections of real disjoint closed intervals. We conclude by providing explicit triangular solutions to constrained Schlesinger systems.
Paper Structure (16 sections, 27 theorems, 139 equations)

This paper contains 16 sections, 27 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Toda curves and Pell equations
  3. Toda families of Riemann surfaces
  4. Meromorphic differentials on Toda curves
  5. Variation of the curve
  6. Rauch variation
  7. Variation of dependent branch points on the Toda families of curves
  8. Variational formulas for the Toda families of curves
  9. Isoperiodicity equations
  10. Generalized Toda families of Riemann surfaces
  11. Isoequilibrium deformations
  12. Deformations of the essential spectra of periodic Jacobi matrices
  13. Isoperiodic deformations of periodic Toda lattices
  14. The Korteweg - de Vries difference equation and Chebyshev polynomials
  15. Applications to the $SU(N)$ Seiberg-Witten theory
  16. ...and 1 more sections

Key Result

Lemma 1

Let $\mathcal{T}_{\bf x}$ be the compact Riemann surface corresponding to the algebraic curve surf and let $P_{a_j}$ denote the ramification point of the covering $u:\mathcal{T}_{\bf x}\to \mathbb CP^1$ corresponding to the branch point $a_j\in \mathcal{B}=\{0,1,x_1, u_1, \dots, x_g, u_g\}.$ Let the

Theorems & Definitions (36)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Theorem 1
  • Remark 1
  • Lemma 2
  • Proposition 1
  • Lemma 3
  • Proposition 2
  • Proposition 3
  • ...and 26 more