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The goal of this paper is to make first steps towards the quantisation ofintegrable non-linear sigma models using the formalism of affine Gaudin models,by approaching these theories through their conformal limits. We focus mostlyon the example of the Klim\v{c}\'{i}k model, which is a two-parameterdeformation of the Principal Chiral Model on a Lie group $G$. We show that theUV fixed point of this theory is described classically by two decoupled chiralaffine Gaudin models, encoding its left- and right-moving degrees of freedom,and give a detailed analysis of the chiral and integrable structures of thesemodels. Their quantisation is then explored within the framework of Feigin andFrenkel. We study the quantum local integrals of motion using the formalism ofquantised affine Gaudin models and show agreement of the first two integralswith known results in the literature for $G={\rm SU}(2)$. Evidence is given forthe existence of a monodromy matrix satisfying the Yang-Baxter algebra for thismodel, thus paving the way for the quantisation of the non-local integrals ofmotion. We conclude with various perspectives, including on generalisations ofthis program to a larger class of integrable sigma models and applications ofthe ODE/IQFT correspondence to the description of their quantum spectrum.

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