Hecke operators and analytic Langlands correspondence for curves over local fields

We explore the difference Langlands correspondence using the four dimensional $$ \mathcal{N} $$ 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$$ \left(\mathfrak{gl}(2)\right) $$ gl 2 , making it the wavefunction of a N-site $$ \mathfrak{gl}(2) $$ gl 2 spin chain with bi-infinite spin modules. We construct the Q- and $$ \overset{\sim }{\textbf{Q}} $$ Q ~ -surface observables which are believed to be the Q-operators on the bi-infinite module over the Yangian Y$$ \left(\mathfrak{gl}(2)\right) $$ gl 2 , and compute their eigenvalues, the Q-functions, as vevs of the surface observables.

di-Langlands correspondence and extended observables

Jeong,

Lee,

Nekrasov

2024

Vertex algebra constructions for (analytic) geometric Langlands in genus zero

Gaiotto

2024

We employ two-dimensional chiral algebra techniques to produce solutions of certain differential and integral equations which occur in the context of the Analytic Geometric Langlands Program. In particular, we build “Multiplication Kernels” K3(x, x′, x′′) which intertwine the action of $$ \mathfrak{s}{\mathfrak{l}}_2 $$ s l 2 Gaudin Hamiltonians on three sets of variables.

Bispectral duality and separation of variables from surface defect transition

Jeong,

Lee

2024

We study two types of surface observables − the Q-observables and the H-observables − of the 4d $$ \mathcal{N} $$ N = 2 A1-quiver U(N) gauge theory obtained by coupling a 2d $$ \mathcal{N} $$ 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($$ \mathfrak{gl} $$ gl (2)) and the Hecke operators on the twisted $$ \hat{\mathfrak{sl}} $$ sl ̂ (N)-coinvariants, we derive an exact duality between the spectral problems of the $$ \mathfrak{gl} $$ gl (2) XXX spin chain with N sites and the $$ \mathfrak{sl} $$ 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 $$ \mathcal{N} $$ 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 $$ \mathfrak{gl} $$ gl (2) XXX spin chain and the $$ \mathfrak{sl} $$ sl (N) Gaudin model, respectively.