{"id":"doi:10.5281/zenodo.23610165","kind":"article","title":"Proof of Quantum Geometric Langlands via 2-Fourier–Mukai Transform — E8 Intelligence Research","authors":["Andrew Stewart Caldin"],"author_count":1,"year":2026,"published":"2026-10-24","venue":"Zenodo (CERN European Organization for Nuclear Research)","cited_by":0,"doi":"10.5281/zenodo.23610165","arxiv_id":null,"nct_id":null,"source_url":"https://doi.org/10.5281/zenodo.23610165","archive_url":"https://space-kitty.com/papers/doi:10.5281/zenodo.23610165","updated":null,"pmid":null,"pmcid":null,"openalex_id":"W7223447952","pdf_url":null,"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_license":"CC-BY (publisher license, via OpenAlex)","license":"cc-by","is_open_access":true,"retracted":false,"peer_reviewed":null,"categories":["Algebraic structures and combinatorial models"],"details":null,"provenance":"openalex","attribution":"Abstract of \"Proof of Quantum Geometric Langlands via 2-Fourier–Mukai Transform — E8 Intelligence Research\" by Andrew Stewart Caldin, https://doi.org/10.5281/zenodo.23610165, used under its Creative Commons Attribution license (CC-BY); markup removed, otherwise unmodified.","topics":[{"slug":"landmark-math-results","name":"Landmark Results in Pure Mathematics","domain_slug":"math-cs"}],"cited_in_dossiers":[]}