Skein-theoretic Langlands duality

Establish whether, for suitable closed oriented 3-manifolds and Langlands-dual groups \(G\) and \(\check G\), the skein theories \(Sk_G(M)\) and \(Sk_{\check G}(M)\) are canonically identified, and develop the corresponding spectral-decomposition interpretation.

Background

The paper formulates a conjectural correspondence between skein theories for Langlands-dual groups. At the time of writing, the cited evidence consists only of dimension comparisons, with no isomorphism constructed. The authors explicitly state that the validity of the proposed correspondence is far from understood.

References

In particular, the deformed theories $\mathcal{A}{G,q}$ and $\mathcal{B}{\check{G}, \check{q}}$ where $q$ and $\check{q}$ are Langlands dual deformation parameters participate in a conjectural skein Langlands duality : for certain closed oriented 3-manifolds $M$, one expects an identification

\mathcal{A}{G,q}(M) = Sk_G(M) \overset{?}{\longleftrightarrow} Sk{\check{G}}(M) = \mathcal{B}_{\check{G}, \check{q}}(M)

analogous to the spectral decomposition of Langlands.

— Skein theory, line defects, and quantum symmetric pairs  (2609.18902 - Chen et al., 16 Sep 2026) in Equation (1.4) and surrounding discussion, Section 1, subsection “Relative Langlands duality”