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On the Integrable Structure of the SU(2) Wess-Zumino-Novikov-Witten Model

Sylvain Lacroix, Adrien Molines

TL;DR

This work establishes a first-principles quantum integrable framework for the SU(2) WZNW model by constructing two distinct towers of local higher-spin integrals of motion (IMs): the KdV-like IMs inside the Virasoro subalgebra and a novel WZNW-intrinsic tower built from Kac–Moody currents that cannot be reduced to the stress tensor. The authors derive explicit forms for the first few local IMs, demonstrate that the new tower commutes with known non-local IMs (Kondo defects), and diagonalise these charges on $ abla\widehat{\mathfrak{su}(2)}_k$ Verma modules, revealing a largely non-degenerate spectrum. They then connect these results to the affine Bethe Ansatz and to the ODE/IQFT correspondence (Langlands duality), providing checks that the complete eigenbasis and eigenvalues can be accessed via these frameworks. The findings support the view that the WZNW integrable structure is governed by a richer algebraic framework than the Virasoro-based KdV structure and offer a pathway toward quantising 2d integrable sigma-models, including massive deformations and higher-rank generalisations. Overall, the paper advances a principled, quantum-field-theoretic avenue for fully characterising the spectrum of 2d integrable CFTs through affine Gaudin-type structures and their ODE/IQFT avatars.

Abstract

This paper is devoted to the quantum integrable structure of Wess-Zumino-Novikov-Witten models, formed by an infinite number of commuting Integrals of Motion (IMs) in their current algebra. Focusing for simplicity on the SU(2) case, we obtain the first four commuting higher-spin local IMs, starting from a general SU(2)-invariant ansatz and imposing their commutativity. We further show evidence of their commutativity with quantum non-local IMs, which were already built in the literature as Kondo defects. We then investigate the diagonalization of these local operators on $\widehat{\mathfrak{su}(2)}_k$ Verma modules: we explicitly find the first few eigenvectors and further discuss the affine Bethe ansatz and ODE/IQFT conjectures, which predict the full eigenstates and spectrum of the integrable structure. Our results show a perfect match between the direct diagonalization and these overarching conjectures. We conclude by discussing several outlooks, including multi-current generalisations, massive deformations and a general long-term program towards the first principle quantisation of 2-dimensional integrable sigma-models.

On the Integrable Structure of the SU(2) Wess-Zumino-Novikov-Witten Model

TL;DR

This work establishes a first-principles quantum integrable framework for the SU(2) WZNW model by constructing two distinct towers of local higher-spin integrals of motion (IMs): the KdV-like IMs inside the Virasoro subalgebra and a novel WZNW-intrinsic tower built from Kac–Moody currents that cannot be reduced to the stress tensor. The authors derive explicit forms for the first few local IMs, demonstrate that the new tower commutes with known non-local IMs (Kondo defects), and diagonalise these charges on Verma modules, revealing a largely non-degenerate spectrum. They then connect these results to the affine Bethe Ansatz and to the ODE/IQFT correspondence (Langlands duality), providing checks that the complete eigenbasis and eigenvalues can be accessed via these frameworks. The findings support the view that the WZNW integrable structure is governed by a richer algebraic framework than the Virasoro-based KdV structure and offer a pathway toward quantising 2d integrable sigma-models, including massive deformations and higher-rank generalisations. Overall, the paper advances a principled, quantum-field-theoretic avenue for fully characterising the spectrum of 2d integrable CFTs through affine Gaudin-type structures and their ODE/IQFT avatars.

Abstract

This paper is devoted to the quantum integrable structure of Wess-Zumino-Novikov-Witten models, formed by an infinite number of commuting Integrals of Motion (IMs) in their current algebra. Focusing for simplicity on the SU(2) case, we obtain the first four commuting higher-spin local IMs, starting from a general SU(2)-invariant ansatz and imposing their commutativity. We further show evidence of their commutativity with quantum non-local IMs, which were already built in the literature as Kondo defects. We then investigate the diagonalization of these local operators on Verma modules: we explicitly find the first few eigenvectors and further discuss the affine Bethe ansatz and ODE/IQFT conjectures, which predict the full eigenstates and spectrum of the integrable structure. Our results show a perfect match between the direct diagonalization and these overarching conjectures. We conclude by discussing several outlooks, including multi-current generalisations, massive deformations and a general long-term program towards the first principle quantisation of 2-dimensional integrable sigma-models.
Paper Structure (100 sections, 155 equations, 1 figure)

This paper contains 100 sections, 155 equations, 1 figure.

Table of Contents

  1. Introduction
  2. First principle quantum integrability.
  3. Integrability in CFT.
  4. Integrable structure of the WZNW model.
  5. Main results for $\bm{G=}\,$SU(2).
  6. Diagonalization of the integrable structure.
  7. Plan of the paper.
  8. 2d CFT and WZNW model on the cylinder
  9. 2d CFT on the cylinder
  10. Chiral algebra and OPE.
  11. Fourier modes and their commutators.
  12. Stress-energy tensor.
  13. Hamiltonian and Hilbert space.
  14. Normal ordering.
  15. Generalised Wick theorem.
  16. ...and 85 more sections

Figures (1)

  • Figure 1: The Hankel contour $\mathcal{C}$.