Integrable sigma models at RG fixed points: quantisation as affine Gaudin models

The goal of my talk will be to summarise recent joint work with G. Kotousov and S. Lacroix approaching the quantisation of some integrable non-linear sigma models through their conformal limits. We focus mostly on the example of the Klimcık model, which is a two-parameter deformation of the Principal Chiral Model on a Lie group G. The UV fixed point of this theory is described classically by two decoupled chiral affine Gaudin models, encoding its left- and right-moving degrees of freedom, respectively. The chiral structure provides the basis for a quantisation of the affine Gaudin models following work of Feigin and Frenkel. The integrable structure of the quantised Klimcık model can be represented either by local or by non-local integrals of motion. The representation as affine Gaudin models allows us to construct quantum local integrals of motion, and suggests a description of their spectra using a variant of the ODE/IQFT-correspondence. Evidence is given for the existence of quantum monodromy matrices satisfying the Yang-Baxter algebra, paving the way for the quantisation of the non-local integrals of motion