Geometric Langlands: Categorical Formulation via Moduli Stacks and Hecke Eigensheaves — E8 Intelligence Research

FINDING: Geometric Langlands correspondence is being actively constructed via moduli stacks of local systems and Hecke eigensheaves, with recent advances by Gaitsgory, Raskin, and others toward a full categorical formulation. | MATH: The correspondence conjecturally identifies the derived category of coherent sheaves on the moduli stack of local systems \( LocSys_G(Σ) \) with the derived category of D-modules on the moduli stack of principal \( G \)-bundles \( Bun_G(Σ) \), for a smooth projective curve \( Σ \). Key objects: Hecke eigensheaves — D-modules \( F \) on \( Bun_G \) satisfying \( H_μ(F) F σ_μ(E) \) for each Hecke correspondence, where \( σ_μ \) is the representation of the Langlands dual group \( {}^L G \) with highest weight \( μ \). The local version (Braverman) involves affine Grassmannian \( Gr_G = G(C((t)))/G(C[[t]]) \) and its c Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com