On the integrable structure of the SU(2) Wess-Zumino-Novikov-Witten model

A bstract 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 $$ {\hat{\mathfrak{su}(2)}}_k $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mrow> <mml:mi>su</mml:mi> <mml:mfenced> <mml:mn>2</mml:mn> </mml:mfenced> </mml:mrow> <mml:mo>̂</mml:mo> </mml:mover> <mml:mi>k</mml:mi> </mml:msub> </mml:math> 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.