Bispectral duality and separation of variables from surface defect transition

TL;DR

This work reveals that transitions between surface defects in 4d $\mathcal{N}=2$ $A_1$-quiver gauge theories, realized via coupling to 2d $\mathcal{N}=(2,2)$ theories, are governed by a Fourier-type transform that interchanges $\mathbf{Q}$- and $\mathbf{H}$-observables. This leads to an exact bispectral duality between the $\mathfrak{gl}(2)$ XXX spin chain (on $N$ bi-infinite modules) and the $\mathfrak{sl}(N)$ Gaudin model (with 4 sites), encoded through the pair of $Q$-operators and Hecke operators acting on twisted coinvariants. The monodromy surface defect admits a dual 2d sigma-model description, and its transitions to multiple $\mathbf{Q}$- or $\mathbf{H}$-observables are implemented by explicit contour-integral transformations, providing $\hbar$-deformed and higher-rank generalizations of the KZ/BPZ correspondence. In the limit $\varepsilon_2\to 0$, these integral transforms yield the quantum separation of variables for both the XXX spin chain and the Gaudin model, with separated variables identified by zeros of appropriate transfer-operator entries. The framework connects gauge theory, Yangians, Hecke operators, and vertex-operator algebras in a unified SoV picture, and points to generalizations to higher-rank quivers and their Chern-Simons/5d avatars.

Abstract

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.

Figures (10)

• Figure 1: Resolution of $\mathbb{C}_{{\varepsilon}_1} \times (\mathbb{C}_{{\varepsilon}_2} \times \mathbb{C}_{{\varepsilon}_4})/\mathbb{Z}_l$ and M5-branes wrapping $\mathbb{C}^\times$.
• Figure 2: IIA brane engineering of the $\mathbf{Q}$-observables, in our case $n=2$; (a) $\mathbf{Q}$-observable and (b) $\tilde{\mathbf{Q}}$-observable (see Table \ref{['table:twM']} and \ref{['table:spacetime']})
• Figure 3: D2-brane creation and higgsing construction of $\mathbf{Q}$-observable. (a) One of $N+1$ D4-branes is aligned across the two NS5-branes, enabling the D4-brane to move along the $x^5$-direction. (b) A D2-brane ending on one of the NS5-branes is created as a result of the transition.
• Figure 4: Transition between $\mathbf{Q}$-observable and $\mathbf{H}$-observable in IIA brane picture. (a) D2-brane meets one of the D4-branes on the left (resp. right) by tuning vev of the complex adjoint scalar (b) D2-brane moves toward left (resp. right) along the D4-brane, turning on positive (resp. negative) real FI parameter.
• Figure 5: D2-brane creation and higgsing construction of the $\mathbf{H}$-observable. (a) All $N$ D4-branes are aligned across one of the NS5-branes, enabling the NS5-brane to move along the $x^5$-direction. (b) A D2-brane ending on one of the D4-branes is created as a result of the higgsing.