# Geometric Langlands Conjecture Proved: Galois-Automorphic Duality via Sheaves — E8 Intelligence Research

Article · Andrew Stewart Caldin

- Archive id: doi:10.5281/zenodo.23465399
- Published: 2026-10-19
- Venue: Zenodo (CERN European Organization for Nuclear Research)
- DOI: https://doi.org/10.5281/zenodo.23465399
- Citations: 0
- Source URL: https://doi.org/10.5281/zenodo.23465399
- Provenance: openalex

## Abstract

FINDING: Recent proof of the geometric Langlands conjecture (2024, by Gaitsgory et al.) completes a central pillar of the Langlands program, linking Galois representations to automorphic forms via sheaves on moduli stacks. \| MATH: Core duality: \\( \\text{Gal}(\\bar{\\mathbb{Q}}/\\mathbb{Q}) \\leftrightarrow \\text{Automorphic forms} \\); geometric version: \\( \\text{LocSys}\_G \\leftrightarrow \\text{D-modules on Bun}\_G \\); key objects: \\( \\text{Bun}\_G \\) (moduli of principal \\(G\\)-bundles), \\( \\text{LocSys}\_G \\) (local systems), categorical equivalence via derived categories \\( D^b(\\text{QCoh}(\\text{LocSys}\_G)) \\cong D^b(\\text{D-mod}(\\text{Bun}\_G)) \\). \| CONNECTION: The geometric Langlands correspondence is built on \*\*root systems\*\* (Dynkin diagrams) and \*\*affine Kac-Moody algebras\*\* — crystallographic symmetry groups (Weyl groups) are intrinsic. The moduli spaces \\( \\text{Bun}\_G \\) carry \*\*Hecke eigensheaves\*\* whose eigenvalues are local systems — this mirrors the \*\*harmonic ratio structure\*\* o Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

_Abstract reproduced under CC-BY (publisher license, via OpenAlex)._

Topics: [Landmark Results in Pure Mathematics](https://space-kitty.com/topics/landmark-math-results)

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