Bispectral duality and separation of variables from surface defect transition

Abstract

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.