Bispectral duality and separation of variables from surface defect transition

We study two types of surface observables $-$ the $\mathbf{Q}$-observables and the $\mathbf{H}$-observables $-$ of the 4d $\mathcal{N}=2$ $A_1$-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 $\mathbf{Q}$-observables and the $\mathbf{H}$-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 $\widehat{\mathfrak{sl}}(N)$-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 $\mathbf{Q}$-observables or $\mathbf{H}$-observables, implemented through integral transformations between their surface observables. These transformations provide, respectively, $\hbar$-deformation and a higher-rank generalization of the KZ/BPZ correspondence. In the limit $\varepsilon_2\to 0$, 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.