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Low-dimensional tori in Calogero-Moser-Sutherland systems

Andrii Liashyk, Guorui Ma, Nicolai Reshetikhin, Ivan Sechin

Abstract

The main result of this paper is an explicit description of the stratification of the phase space of Calogero--Moser--Sutherland (CMS) integrable systems corresponding to Lie groups $SU(n)$. The phase space decomposes into symplectic strata of dimensions $2s$, where $s = 0, 1, \ldots, n - 1$. On each stratum of the positive dimension, we construct natural action-angle coordinates and compute the symplectic form explicitly, showing that every stratum is symplectomorphic to $\mathbb{R}_{> 0}^s \times \mathbb{T}^s$. The zero-dimensional stratum corresponds to the equilibrium point of the multi-time CMS dynamics.

Low-dimensional tori in Calogero-Moser-Sutherland systems

Abstract

The main result of this paper is an explicit description of the stratification of the phase space of Calogero--Moser--Sutherland (CMS) integrable systems corresponding to Lie groups . The phase space decomposes into symplectic strata of dimensions , where . On each stratum of the positive dimension, we construct natural action-angle coordinates and compute the symplectic form explicitly, showing that every stratum is symplectomorphic to . The zero-dimensional stratum corresponds to the equilibrium point of the multi-time CMS dynamics.

Paper Structure

This paper contains 25 sections, 11 theorems, 127 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Calogero--Moser--Sutherland system
  3. Hamiltonian reduction along the coadjoint orbit
  4. Minimal nontrivial coadjoint orbit and CMS phase space
  5. The diagonalization of $g$ and natural coordinates
  6. Lax matrix and Hamiltonians
  7. Base of the Lagrangian projection for the CMS model and stratification of $\mathcal{S}(\mathcal{O})$
  8. CMS system as a Lagrangian fibration
  9. Base of the Lagrangian projection
  10. Description of strata of $\mathcal{S}(\mathcal{O})$
  11. The maximal stratum
  12. Low-dimensional strata $\mathcal{S}(\mathcal{O})_{\{\mathbf{j}\}}$
  13. The dynamical stratification of the phase space
  14. Action-angle variables on $\mathcal{S}(\mathcal{O})_{\{\mathbf{j}\}}$
  15. Natural coordinates on $\mathcal{S}(\mathcal{O})_{\{\mathbf{j}\}}$
  16. ...and 10 more sections

Key Result

Lemma 1

Let $D$ be a diagonal $n \times n$ matrix with entries $D_{ab} = D_a \delta_{ab}$ and let $P$ be a rank-one matrix of the form $P_{ab} = u_a v_b$. Then

Figures (1)

  • Figure 1: The shifted principal Weyl chamber for $G = SU(3)$ and dimensions of Liouville tori over the corresponding strata.

Theorems & Definitions (22)

  • Lemma 1
  • Theorem 1
  • proof
  • Corollary 1
  • Definition 1
  • Corollary 2
  • Definition 2
  • Lemma 2
  • Theorem 2
  • proof
  • ...and 12 more