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

TL;DR

This paper explores the quantum integrable structure of the SU(2) Wess-Zumino-Novikov-Witten model, deriving higher-spin integrals of motion, analyzing their eigenstates, and confirming conjectures about the spectrum and eigenvectors.

Contribution

It explicitly constructs the first four local higher-spin integrals of motion for SU(2) WZNW models and verifies their eigenstates align with the affine Bethe ansatz and ODE/IQFT conjectures.

Findings

Derived the first four commuting higher-spin local IMs.

Found eigenvectors and eigenvalues consistent with Bethe ansatz predictions.

Provided evidence for the commutativity of local and non-local IMs.

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.