di-Langlands correspondence and extended observables

TL;DR

This work presents a concrete gauge-theoretic realization of the di-Langlands (difference Langlands) program using four-dimensional ${\EuScript{N}}=2$ SQCD with surface defects. It constructs the full quantum integrals of motion for a bi-infinite ${\mathfrak{gl}(2)}$ spin-chain realization, identifies the regular monodromy surface defect with a basis of a Yangian ${Y}(\mathfrak{gl}(2))$-module, and shows that ${\bf Q}$- and ${\tilde{\bf Q}}$-observables act as $Q$-operators whose eigenvalues are the $Q$-functions solving ${\hbar}$-difference opers (quantum Baxter TQ equations). The analysis yields a consistent framework in which the eigenstates of the $Q$-operators are simultaneously eigenstates of the XXX spin-chain Hamiltonians, thereby linking gauge-theoretic correlators to integrable systems and to (quantum) opers, providing an $\,\hbar$-deformation of geometric Langlands. The work also outlines rich future directions, including 5d uplifts, affinization, and connections to quantum vertex algebras and analytic Langlands. Overall, it foregrounds surface defects as the central tools for realizing the spectral data and dualities at the heart of the di-Langlands correspondence.

Abstract

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.