Quantum Geometric Langlands via Fundamental Local Equivalence — E8 Intelligence Research

FINDING: Gaitsgory–Lurie's "fundamental local equivalence" replaces Satake equivalence in quantum geometric Langlands, proven for most cases by Dhillon et al. (arXiv:1907.03204), extending to affine flag varieties. MATH: - Core object: categorical equivalence between factorization modules (or D-modules on loop groups) and representations of the quantum group / Kac-Moody algebra at critical level. - Key structure: **quantum parameter** \( \psi \) (level) — the equivalence holds for generic \( \psi \), degenerating at roots of unity where quantum groups become finite-dimensional. - The classical Satake equivalence: \( \text{Rep}(\check{G}) \cong \text{Perv}(\text{Gr}_G) \) — replaced by a **local equivalence** involving **factorization algebras** and **affine flag varieties** (not just Grassmannians). - No explicit numerical constants appear in the abstract; the mathematics is categorical (∞-categories, ind-coherent sheaves, chiral algebras). CONNECTION: - **Root systems*

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