Quantum Geometric Langlands: A Categorical Equivalence via Fundamental Local Duality — E8 Intelligence Research

FINDING: The quantum geometric Langlands program centers on a categorical equivalence between derived categories of sheaves on moduli stacks of G-bundles and those on dual Langlands group ^G-bundles, with the "fundamental local equivalence" as the core new structural result (2024–2026 talks by Gaitsgory, Dhillon, Hayash). | MATH: The equivalence is conjecturally of the form: \( \mathrm{QCoh}(\mathrm{LocSys}_{^G G}(X)) \cong \mathrm{IndCoh}(\mathrm{Bun}_G(X)) \) twisted by a quantum parameter \(\hbar\). The "fundamental local equivalence" refines this to a local statement: for a formal disk \(D\), the category of factorization modules over the affine Grassmannian \(\mathrm{Gr}_G\) with level \(\hbar\) is equivalent to that for \(^G G\) with level \(1/\hbar\) — i.e., \(\mathrm{FactMod}_{\hbar}(\mathrm{Gr}_G) \simeq \mathrm{FactMod}_{1/\hbar}(\mathrm{Gr}_{^G G})\). This is a categorical Fourier–Mukai transform, with the duality parameter \(\hbar \leftrightarrow 1/\hbar\) mirroring \(q \le

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com