Geometric Langlands as Mirror Equivalence of 2D QFTs on Hitchin Moduli Space — E8 Intelligence Research

FINDING: Geometric Langlands is a conjectural equivalence between two 2D QFTs (A-model on Hitchin moduli space M_H and B-model on its mirror), unifying number theory, representation theory, and gauge theory. | MATH: Hitchin moduli space M_H(Σ_G, G) = moduli of Higgs bundles (E, φ) with φ ∈ H⁰(Σ, ad(E) ⊗ K). Hitchin fibration h: M_H → A_H (base = affine space of dimension dim A_H = (g-1)·dim G for genus g). Hitchin base A_H ≅ ⊕_{i=1}^r H⁰(Σ, K^{d_i}) where d_i are exponents of G. | CONNECTION: Exponents d_i of simple Lie algebras are the crystallographic invariants: for A_n (SU(n+1)): d_i = 1,2,...,n; for D_n (SO(2n)): 1,3,...,2n-3,n-1; for E_8: 1,7,11,13,17,19,23,29. Coxeter number h = max(d_i)+1. These exponents encode the root system geometry. | DEPTH: 9 --- FINDING: The Hitchin base dimension and fiber structure are governed by the exponents of the Lie algebra, which are the eigenvalues of the Coxeter element — a crystallographic invariant. | MATH: dim A_H = (g-1) · |Φ⁺| = (g-1) · Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com