Kähler Bands Unify Berry Curvature and Quantum Metric in Chern Insulators — E8 Intelligence Research

FINDING: Kähler bands unify Chern insulator topology with quantum geometry via holomorphic line bundles — Berry curvature and quantum metric are not independent but are two components of a single Hermitian structure on the band manifold. MATH: - Quantum geometric tensor: Q_ij = g_ij + (i/2) F_ij, where g_ij = Re(Q_ij) is the quantum metric and F_ij = Im(Q_ij) is the Berry curvature (2-form). - Kähler compatibility condition: g(JX, JY) = g(X, Y) and ω(X, Y) = g(JX, Y), with dω = 0 (closed Kähler form) and J integrable (Nijenhuis vanishing). - For a holomorphic line bundle L → M with Hermitian metric h, the Chern connection gives F = -∂∂̄ log h (Chern curvature = Berry curvature in the band context). - First Chern number: c₁ = (1/2π) ∫ F ∈ ℤ — quantized Berry phase. - Kähler condition on bands: Berry curvature F is the Kähler form ω; quantum metric g is the Kähler metric; complex structure J is induced by the holomorphic band framing. - Fubini–Study metric on CP^{n-1} (relevant for mul Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com