# Proof of Quantum Geometric Langlands via 2-Fourier–Mukai Transform — E8 Intelligence Research

Article · Andrew Stewart Caldin

- Archive id: doi:10.5281/zenodo.23610165
- Published: 2026-10-24
- Venue: Zenodo (CERN European Organization for Nuclear Research)
- DOI: https://doi.org/10.5281/zenodo.23610165
- Citations: 0
- Source URL: https://doi.org/10.5281/zenodo.23610165
- Provenance: openalex

## Abstract

FINDING: The "fundamental local equivalence" in quantum geometric Langlands is a conjectural categorical substitute for the Satake equivalence, now proven in most cases (Gaitsgory–Lurie conjecture, extended to affine flag varieties by Dhillon et al.), via a 2-Fourier–Mukai transform for non-constant group schemes. MATH: The core object is a categorical equivalence between factorization modules (or D-modules on the affine Grassmannian/flag variety) and representations of the Langlands dual group, twisted by a quantum parameter. Key structures: - \*\*Satake equivalence\*\* (classical): \\( \\text{Rep}(\\check{G}) \\cong \\text{Perv}\_{G(\\mathcal{O})}(\\text{Gr}\_G) \\) — replaced by a \*fundamental local equivalence\*: \\( \\text{QCoh}(\\text{LocSys}\_{\\check{G}})^{\\text{fact}} \\cong \\text{IndCoh}(\\text{Gr}\_G)^{\\text{fact}} \\) (or similar), with a \*\*2-Fourier–Mukai kernel\*\* implementing the transform. - The quantum parameter \\( \\psi \\) (level) enters as a twist; the equivalence is expected to be \*\*se 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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