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Isotropic embeddings of coadjoint orbits and magnetic geodesic flows

Dmitri Bykov, Andrew Kuzovchikov

TL;DR

The paper develops a framework for isotropic and, where possible, Lagrangian embeddings of coadjoint orbits (generalized flag manifolds) into products of coadjoint orbits, extending known SU($n$) results to SO and Sp cases. It constructs embeddings via decompositions $\Uplambda=\sum_i C_i\Uplambda_i$ and analyzes the corresponding moment-map constraints to obtain isotropy and, in favorable instances, Lagrangian submanifolds, including an exceptional Lagrangian series. The second part applies these geometric embeddings to dynamical systems with SU($n$) symmetry, proving an equivalence between magnetic geodesic flows on flag manifolds and certain classical spin-chain models (a Haldane-type mapping), and it discusses quantization from a geometric-quantization perspective. The work provides explicit magnetic geodesic constructions for $\mathbb{CP}^1$ and higher flag manifolds, illustrating how symplectic and group-theoretic structures underpin the duality between sigma-model–type dynamics and spin-chain dynamics, with broad implications for representation theory and mathematical physics.

Abstract

We consider isotropic and Lagrangian embeddings of coadjoint orbits of compact Lie groups into products of coadjoint orbits. After reviewing the known facts in the case of $\mathrm{SU}(n)$ we initiate a similar study for $\mathrm{SO}$ and $\mathrm{Sp}$ cases. In the second part we apply this to the study of dynamical systems with $\mathrm{SU}(n)$ symmetry, proving equivalence between systems of two types: those describing magnetic geodesic flow on flag manifolds and classical `spin chains' of a special type.

Isotropic embeddings of coadjoint orbits and magnetic geodesic flows

TL;DR

The paper develops a framework for isotropic and, where possible, Lagrangian embeddings of coadjoint orbits (generalized flag manifolds) into products of coadjoint orbits, extending known SU() results to SO and Sp cases. It constructs embeddings via decompositions and analyzes the corresponding moment-map constraints to obtain isotropy and, in favorable instances, Lagrangian submanifolds, including an exceptional Lagrangian series. The second part applies these geometric embeddings to dynamical systems with SU() symmetry, proving an equivalence between magnetic geodesic flows on flag manifolds and certain classical spin-chain models (a Haldane-type mapping), and it discusses quantization from a geometric-quantization perspective. The work provides explicit magnetic geodesic constructions for and higher flag manifolds, illustrating how symplectic and group-theoretic structures underpin the duality between sigma-model–type dynamics and spin-chain dynamics, with broad implications for representation theory and mathematical physics.

Abstract

We consider isotropic and Lagrangian embeddings of coadjoint orbits of compact Lie groups into products of coadjoint orbits. After reviewing the known facts in the case of we initiate a similar study for and cases. In the second part we apply this to the study of dynamical systems with symmetry, proving equivalence between systems of two types: those describing magnetic geodesic flow on flag manifolds and classical `spin chains' of a special type.

Paper Structure

This paper contains 16 sections, 9 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Generalized flag manifolds
  3. Isotropic and Lagrangian embeddings.
  4. $\mathrm{SU}(n)$ case
  5. Geodesic flow on $\mathds{CP}^1$.
  6. $\mathrm{SO}, \mathrm{Sp}$ cases: examples
  7. $\mathrm{SO}(2n+1)$ and $\mathrm{Sp}(n)$.
  8. An exceptional Lagrangian series.
  9. General embeddings $\mu=0$
  10. $\mathrm{SU}(n)$ flag manifolds.
  11. Embeddings of generalized flag manifolds.
  12. Quantization.
  13. Magnetic geodesics
  14. Magnetic geodesics on $\mathds{CP}^1$.
  15. Magnetic geodesics on $\mathcal{F}_3$ and $\mathcal{F}_n$.
  16. ...and 1 more sections

Key Result

Proposition 1

Under the conditions (Lambda1), (Lambdasum), (LambdaStab), $\mathcal{O}_{\Uplambda}^{\,\mathrm{G}}$ is an isotropic submanifold of $\prod\limits_{i=0}^r\mathcal{O}_{\Uplambda_i}^{\,\mathrm{G}}$, embedded via the map (orbitmap).

Theorems & Definitions (16)

  • Proposition 1
  • Corollary 1
  • proof
  • Proposition 2: Bykov:2012am
  • Proposition 3: Bykov_2024
  • proof
  • Example 1
  • Proposition 4
  • proof
  • Lemma 1
  • ...and 6 more