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Manifolds of vortex loops as coadjoint orbits

Ioana Ciuclea, Cornelia Vizman

TL;DR

The paper studies coadjoint orbits of the area-preserving diffeomorphism group $\operatorname{Ham}_c(\mathbb{R}^2)$ consisting of vortex loops, i.e., closed plane curves endowed with vorticity densities that may vanish. It builds a geometric framework by modeling vortex loops as embeddings $\operatorname{Emb}_a(S^1,\mathbb{R}^2)$ equipped with a hat-calculus symplectic form $\Omega$, and then analyzes how rotational symmetry and Morse zeros affect the orbit structure, yielding identifications with quotient spaces $\operatorname{Emb}_a(S^1,\mathbb{R}^2)/\mathbb{Z}_{k/\ell}$ and the associated Kirillov-Kostant-Souriau (KKS) symplectic forms. The work extends to pointed vortex loops via a combined form $\Omega^\Gamma$ when point vortices are attached, showing that the same orbit can be realized via either zeros of the vorticity density or discrete point vortices, and clarifying when each realization yields a descended KKS form. It thereby integrates two viewpoints—zeros of a Morse one-form and attached point vortices—into a unified coadjoint-orbit description, with a decorated Grassmannian interpretation and explicit momentum maps connecting the embedding space to the dual of the Lie algebra of Hamiltonian vector fields.

Abstract

We study a class of coadjoint orbits of the area preserving diffeomorphism group of the plane consisting of vortex loops, namely closed curves in the plane decorated with one-forms (vorticity densities) allowed to have zeros.

Manifolds of vortex loops as coadjoint orbits

TL;DR

The paper studies coadjoint orbits of the area-preserving diffeomorphism group consisting of vortex loops, i.e., closed plane curves endowed with vorticity densities that may vanish. It builds a geometric framework by modeling vortex loops as embeddings equipped with a hat-calculus symplectic form , and then analyzes how rotational symmetry and Morse zeros affect the orbit structure, yielding identifications with quotient spaces and the associated Kirillov-Kostant-Souriau (KKS) symplectic forms. The work extends to pointed vortex loops via a combined form when point vortices are attached, showing that the same orbit can be realized via either zeros of the vorticity density or discrete point vortices, and clarifying when each realization yields a descended KKS form. It thereby integrates two viewpoints—zeros of a Morse one-form and attached point vortices—into a unified coadjoint-orbit description, with a decorated Grassmannian interpretation and explicit momentum maps connecting the embedding space to the dual of the Lie algebra of Hamiltonian vector fields.

Abstract

We study a class of coadjoint orbits of the area preserving diffeomorphism group of the plane consisting of vortex loops, namely closed curves in the plane decorated with one-forms (vorticity densities) allowed to have zeros.
Paper Structure (4 sections, 10 theorems, 31 equations)

This paper contains 4 sections, 10 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Manifolds of vortex loops
  3. Symplectic structures for manifolds of embeddings
  4. Coadjoint orbits of the Hamiltonian group of $\mathbb R^2$

Key Result

Lemma 2.4

The group homomorphism where $j$ is a natural number such that $\gamma(t_i) = t_{i+j}$, as described in Remark def_j, is injective. Its image is the subgroup of $\mathbb Z_k$ generated by the element $\ell \mod k$ and is isomorphic to the cyclic group of degree $k/\ell$, namely $\mathbb Z_{k/\ell}$.

Theorems & Definitions (15)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Corollary 2.6
  • Remark 2.7: Decorated nonlinear Grassmannians framework HV_decorated
  • Lemma 3.1
  • Proposition 3.2
  • Proposition 4.1
  • ...and 5 more