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Sigma models from Gaudin spin chains

Dmitri Bykov, Andrew Kuzovchikov

TL;DR

This work establishes a rigorous bridge between one-dimensional sigma models with flag-manifold targets and SU$(N)$ spin chains of Gaudin type. By mapping classical geodesic flow and quantum Laplace-Beltrami spectra to Gaudin integrable systems, it provides explicit geodesics on ${\mathcal F}(3)$ in terms of elliptic functions and Lamé equations, and determines the Laplacian spectrum via Bethe ansatz and Heun polynomials. A complementary spectral reconstruction method leverages $S_3$ symmetry to infer parts of the spectrum without Bethe ansatz, offering cross-checks and insights for small $p$. The results generalize to ${\mathcal F}(N)$ and partial flag manifolds, with potential relevance to AdS$_4$ compactifications and magnetic-geodesic generalizations, highlighting a versatile framework for classical-quantum correspondence in flag-manifold settings.

Abstract

We solve the classical and quantum problems for the 1D sigma model with target space the flag manifold $\mathrm{U}(3)\over \mathrm{U}(1)^3$, equipped with the most general invariant metric. In particular, we explicitly describe all geodesics in terms of elliptic functions and demonstrate that the spectrum of the Laplace-Beltrami operator may be found by solving polynomial (Bethe) equations. The main technical tool that we use is a mapping between the sigma model and a Gaudin model, which is also shown to hold in the $\mathrm{U}(n)$ case.

Sigma models from Gaudin spin chains

TL;DR

This work establishes a rigorous bridge between one-dimensional sigma models with flag-manifold targets and SU spin chains of Gaudin type. By mapping classical geodesic flow and quantum Laplace-Beltrami spectra to Gaudin integrable systems, it provides explicit geodesics on in terms of elliptic functions and Lamé equations, and determines the Laplacian spectrum via Bethe ansatz and Heun polynomials. A complementary spectral reconstruction method leverages symmetry to infer parts of the spectrum without Bethe ansatz, offering cross-checks and insights for small . The results generalize to and partial flag manifolds, with potential relevance to AdS compactifications and magnetic-geodesic generalizations, highlighting a versatile framework for classical-quantum correspondence in flag-manifold settings.

Abstract

We solve the classical and quantum problems for the 1D sigma model with target space the flag manifold , equipped with the most general invariant metric. In particular, we explicitly describe all geodesics in terms of elliptic functions and demonstrate that the spectrum of the Laplace-Beltrami operator may be found by solving polynomial (Bethe) equations. The main technical tool that we use is a mapping between the sigma model and a Gaudin model, which is also shown to hold in the case.

Paper Structure

This paper contains 20 sections, 10 theorems, 102 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Flag manifolds
  3. $\mathrm{SU}(N)$ spin chains and full flag manifolds
  4. Geodesics on $\mathcal{F}(3)$
  5. Integrals of motion.
  6. Elliptic curve and the $\wp$-function.
  7. Lamé equation and its solution.
  8. Special solutions.
  9. Geodesics in $\mathds{CP}^1$.
  10. Other special solutions.
  11. Spectrum of the Laplacian on $\mathcal{F}(3)$
  12. Functions on $\mathcal{F}(3)$ and explicit computation of the spectrum.
  13. Bethe ansatz and Heun polynomials.
  14. Spectral reconstruction
  15. Conclusion and outlook
  16. ...and 5 more sections

Key Result

Proposition 1

$\mathcal{F}_{n_1,\dots,n_k}$ is a Lagrangian submanifold of $(\mathrm{Gr}(n_1,N)\times\dots\times\mathrm{Gr}(n_k,N), \Omega)$, where the symplectic form $\Omega$ is a sum of Fubini-Study forms and the submanifold is distinguished by the orthogonality condition on the planes corresponding to points

Figures (4)

  • Figure 1: Decomposition of $\mathrm{L}^2(\mathcal{F}(3))$ into irreducible representations. Integers inside the table stand for multiplicities of the corresponding representations.
  • Figure 2: Representations for which the spectrum can be explicitly determined using spectral reconstruction (the hook between the blue axes and the red lines).
  • Figure 3: Roots of the polynomials.
  • Figure 4: Degenerate case.

Theorems & Definitions (12)

  • Proposition 1: BykLagEmb
  • Proposition 2: Bykov_2024
  • Proposition 3: Bykov_2024
  • Lemma 1
  • proof
  • Proposition 4: MiFom
  • Proposition 5
  • Lemma 2
  • proof
  • Proposition 6
  • ...and 2 more