Quantum Geometric Langlands: A Categorical Equivalence via Fundamental Local Duality — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The quantum geometric Langlands program centers on a categorical equivalence between derived categories of sheaves on moduli stacks of G-bundles and those on dual Langlands group ^G-bundles, with the "fundamental local equivalence" as the core new structural result (2024–2026 talks by Gaitsgory, Dhillon, Hayash). | MATH: The equivalence is conjecturally of the form: QCoh(LocSysG G(X)) ≅ IndCoh(BunG(X)) twisted by a quantum parameter ℏ. The "fundamental local equivalence" refines this to a local statement: for a formal disk D, the category of factorization modules over the affine Grassmannian GrG with level ℏ is equivalent to that for G G with level 1/ℏ — i.e., FactModℏ(GrG) ≃ FactMod1/ℏ(GrG G). This is a categorical Fourier–Mukai transform, with the duality parameter ℏ ↔ 1/ℏ mirroring \(q \le
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026quantum,
title = {{Quantum Geometric Langlands: A Categorical Equivalence via Fundamental Local Duality — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23526294},
url = {https://zenodo.org/records/23526294},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantum Geometric Langlands: A Categorical Equivalence via Fundamental Local Duality — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/21
PB - Zenodo
DO - 10.5281/zenodo.23526294
UR - https://zenodo.org/records/23526294
AB - FINDING: The quantum geometric Langlands program centers on a categorical equivalence between derived categories of sheaves on moduli stacks of G-bundles and those on dual Langlands group ^G-bundles, with the "fundamental local equivalence" as the core new structural result (2024–2026 talks by Gaitsgory, Dhillon, Hayash). | MATH: The equivalence is conjecturally of the form: QCoh(LocSys_^G G(X)) ≅ IndCoh(Bun_G(X)) twisted by a quantum parameter ℏ. The "fundamental local equivalence" refines this to a local statement: for a formal disk D, the category of factorization modules over the affine Grassmannian Gr_G with level ℏ is equivalent to that for ^G G with level 1/ℏ — i.e., FactMod_ℏ(Gr_G) ≃ FactMod_1/ℏ(Gr_^G G). This is a categorical Fourier–Mukai transform, with the duality parameter ℏ ↔ 1/ℏ mirroring \(q \le Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -