Quantum Geometric Langlands Duality Linked to Symplectic Duality and Affine Weyl Structure — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Quantum geometric Langlands duality is fundamentally tied to symplectic duality and level-shift by the dual Coxeter number, with affine Weyl group structure governing the self-duality condition. | MATH: The key structure is the quantum geometric Langlands correspondence: for a reductive group G and its Langlands dual L G, the duality relates G-modules at level k to L G-modules at level k' where k' = 1/k (up to shift). The critical shift is k → k + h∨ where h∨ is the dual Coxeter number. Self-duality occurs when k = ± √(h∨ L h∨) or more precisely when the level satisfies k2 = h∨ L h∨ (for simply-laced, h∨ = L h∨, giving k = h∨). The affine Weyl group W̃ acts on the level space, and the duality is an involution generated by the affine Weyl group element s0 (the affine simple reflection). The "quantum" parameter \(q = e^{2\pi i/(k+h^\ve
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 Duality Linked to Symplectic Duality and Affine Weyl Structure — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635303},
url = {https://zenodo.org/records/23635303},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantum Geometric Langlands Duality Linked to Symplectic Duality and Affine Weyl Structure — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635303
UR - https://zenodo.org/records/23635303
AB - FINDING: Quantum geometric Langlands duality is fundamentally tied to symplectic duality and level-shift by the dual Coxeter number, with affine Weyl group structure governing the self-duality condition. | MATH: The key structure is the quantum geometric Langlands correspondence: for a reductive group G and its Langlands dual ^L G, the duality relates G-modules at level k to ^L G-modules at level k' where k' = 1/k (up to shift). The critical shift is k → k + h^∨ where h^∨ is the dual Coxeter number. Self-duality occurs when k = ± √(h^∨ ^L h^∨) or more precisely when the level satisfies k^2 = h^∨ ^L h^∨ (for simply-laced, h^∨ = ^L h^∨, giving k = h^∨). The affine Weyl group W̃ acts on the level space, and the duality is an involution generated by the affine Weyl group element s_0 (the affine simple reflection). The "quantum" parameter \(q = e^{2\pi i/(k+h^\ve Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -