We study two types of surface observables − the Q-observables and the H-observables − of the 4d \( \mathcal{N} \) = 2 A1-quiver U(N) gauge theory obtained by coupling a 2d \( \mathcal{N} \) = (2, 2) gauged linear sigma model. We demonstrate that the transition between the two surface defects manifests as a Fourier transformation between the surface observables. Utilizing the results from our previous works, which establish that the Q-observables and the H-observables give rise, respectively, to the Q-operators on the evaluation module over the Yangian Y( \( \mathfrak{gl} \) (2)) and the Hecke operators on the twisted \( \hat{\mathfrak{sl}} \) (N)-coinvariants, we derive an exact duality between the spectral problems of the \( \mathfrak{gl} \) (2) XXX spin chain with N sites and the \( \mathfrak{sl} \) (N) Gaudin model with 4 sites, both of which are defined on bi-infinite modules. Moreover, we present a dual description of the monodromy surface defect as coupling a 2d \( \mathcal{N} \) = (2, 2) gauged linear sigma model. Employing this dual perspective, we demonstrate how the monodromy surface defect undergoes a transition to multiple Q-observables or H-observables, implemented through integral transformations between their surface observables. These transformations provide, respectively, ħ-deformation and a higher-rank generalization of the KZ/BPZ correspondence. In the limit ε2 → 0, they give rise to the quantum separation of variables for the \( \mathfrak{gl} \) (2) XXX spin chain and the \( \mathfrak{sl} \) (N) Gaudin model, respectively.
