Kähler Structure Unifies Quantum Metric and Berry Curvature in Chern Bands — E8 Intelligence Research

FINDING: Chern-insulator Bloch bands admit a Kähler structure — the quantum metric and Berry curvature are the real and imaginary parts of a single Hermitian (Kähler) metric on the band's holomorphic line bundle, unifying quantum geometry into one complex object. MATH: - Quantum geometric tensor (Provost–Vallee, 1980): Q_μν = ⟨∂_μ u | (1 − P) | ∂_ν u⟩ = g_μν + (i/2) Ω_μν - g_μν = Re Q_μν (Fubini–Study / quantum metric, symmetric) - Ω_μν = −2 Im Q_μν (Berry curvature, antisymmetric) - Fubini–Study metric on CP^{N−1}: ds² = (⟨du|du⟩⟨u|u⟩ − |⟨u|du⟩|²)/⟨u|u⟩² - Berry curvature integral (Chern number): C = (1/2π) ∮_BZ Ω d²k ∈ ℤ - Kähler condition (Mera et al.): for a holomorphic band, g and Ω satisfy the Kähler compatibility ∇_μ Ω_νλ = ∇_ν Ω_μλ, with Ω as the Kähler form and g as the Kähler metric. - Trace identity: tr g = |Ω| in the "ideal" (lowest-Landau-level-like) flat-band limit — the saturation condition g_μν = (|Ω|/2) δ_μν. CONNECTION: - The Kähler form Ω is a symplectic 2-form Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com