Langlands Duality: Root System Transposition, Symmetry Exchange, and Topological Mirror Extensions — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Langlands duality maps a root system to its dual via transposition of the Cartan matrix, exchanging long/short roots and Weyl group orbits; topological and mirror-symmetry incarnations extend this to 1-form symmetries and Higgs bundle moduli. | MATH: For simple Lie algebra 𝔤 with root system Φ and coroots Φ∨, the Langlands dual 𝔤∨ has roots Φ∨ (as vectors in dual space). Cartan matrix A_ij = 2(α_i,α_j)/(α_j,α_j) → A^T_ij for 𝔤∨. Weyl group W(Φ) ≅ W(Φ∨) (isomorphic as abstract groups) but acts on dual lattices. Killing form scales: (·,·)∨ = (·,·)/c where c is the ratio of squared lengths (e.g., for SO(2n+1) vs Sp(2n): c=2, so long roots of SO become short of Sp). | CONNECTION: Root systems are crystallographic — their Weyl groups are finite reflection groups with Coxeter numbers h. For SO(2n+1) (B_n) and Sp(2n) (C_n), h = 2n. The duality exchanges root lengths: ratio of squared lengths = 2, which appears in the golden-ratio-adjacent context (2 = φ + 1/φ ≈ 2.236? No — exact 2, n
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026langlands,
title = {{Langlands Duality: Root System Transposition, Symmetry Exchange, and Topological Mirror Extensions — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23526078},
url = {https://zenodo.org/records/23526078},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Langlands Duality: Root System Transposition, Symmetry Exchange, and Topological Mirror Extensions — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/21
PB - Zenodo
DO - 10.5281/zenodo.23526078
UR - https://zenodo.org/records/23526078
AB - FINDING: Langlands duality maps a root system to its dual via transposition of the Cartan matrix, exchanging long/short roots and Weyl group orbits; topological and mirror-symmetry incarnations extend this to 1-form symmetries and Higgs bundle moduli. | MATH: For simple Lie algebra 𝔤 with root system Φ and coroots Φ∨, the Langlands dual 𝔤∨ has roots Φ∨ (as vectors in dual space). Cartan matrix A_ij = 2(α_i,α_j)/(α_j,α_j) → A^T_ij for 𝔤∨. Weyl group W(Φ) ≅ W(Φ∨) (isomorphic as abstract groups) but acts on dual lattices. Killing form scales: (·,·)∨ = (·,·)/c where c is the ratio of squared lengths (e.g., for SO(2n+1) vs Sp(2n): c=2, so long roots of SO become short of Sp). | CONNECTION: Root systems are crystallographic — their Weyl groups are finite reflection groups with Coxeter numbers h. For SO(2n+1) (B_n) and Sp(2n) (C_n), h = 2n. The duality exchanges root lengths: ratio of squared lengths = 2, which appears in the golden-ratio-adjacent context (2 = φ + 1/φ ≈ 2.236? No — exact 2, n Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -