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 \( \mathrm{LocSys}_G(\Sigma) \) with the derived category of D-modules on the moduli stack of principal \( G \)-bundles \( \mathrm{Bun}_G(\Sigma) \), for a smooth projective curve \( \Sigma \). Key objects: Hecke eigensheaves — D-modules \( \mathcal{F} \) on \( \mathrm{Bun}_G \) satisfying \( H_\mu(\mathcal{F}) \cong \mathcal{F} \boxtimes \sigma_\mu(\mathcal{E}) \) for each Hecke correspondence, where \( \sigma_\mu \) is the representation of the Langlands dual group \( {}^L G \) with highest weight \( \mu \). The local version (Braverman) involves affine Grassmannian \( \mathrm{Gr}_G = G(\mathbb{C}((t)))/G(\mathbb{C}[[t]]) \) and its c

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