di-Langlands correspondence and extended observables

We explore the $\textit{difference Langlands correspondence}$ using the four dimensional ${\mathcal{N}}=2$ super-QCD. Surface defects and surface observables play the crucial role. As an application, we give the first construction of the full set of quantum integrals, i.e. commuting differential operators, such that the partition function of the so-called regular monodromy surface defect is their joint eigenvectors in an evaluation module over the Yangian $Y(\mathfrak{gl}(2))$, making it the wavefunction of a $N$-site $\mathfrak{gl}(2)$ spin chain with bi-infinite spin modules. We construct the $\mathbf{Q}$- and $\tilde{\mathbf{Q}}$-surface observables which are believed to be the $Q$-operators on the bi-infinite module over the Yangian $Y(\mathfrak{gl}(2))$, and compute their eigenvalues, the $Q$-functions, as vevs of the surface observables.