Define a new q-deformation of the KZ equation for the non-simply laced two-parameter Langlands correspondence

Define a new q-deformation of the Knizhnik–Zamolodchikov equation for the affine Kac–Moody algebra \widehat{^L g} at level ^L k that replaces the usual qKZ equations associated with U_\hbar(\widehat{^L g})_{^L k} on the electric side of the two-parameter Langlands correspondence when g is non-simply laced.

Background

The paper establishes a canonical identification of deformed conformal blocks for simply-laced Lie algebras between the quantum affine algebra U_\hbar(\widehat{^L g}){^L k} and the deformed W-algebra \mathcal{W}{q,t}(g), with parameters related by explicit formulas. In the non-simply laced case, the authors show that the critical-level limit of the usual qKZ equations on the electric side yields eigenvectors of the XXZ model for U_t(^L\widehat{g}), whereas the magnetic side degenerates to \mathcal{W}_{1,t}(g), which corresponds instead to the folded quantum integrable model. This mismatch implies that the standard qKZ system cannot govern the non-simply laced correspondence.

Consequently, the authors conclude that a different system of q-difference equations is required on the electric side in the non-simply laced setting: specifically, a new q-deformation of the Knizhnik–Zamolodchikov equation for \widehat{^L g} at level ^L k. They note that this deformation has not yet been defined and indicate that they will address it in forthcoming work.