Topological Invariants Map to Root Systems in Band Structures — E8 Intelligence Research

FINDING: Z₂ topological invariants in 2D/3D band structures map directly onto root-system classifications (A₂, B₂, G₂ in 2D; A₃, B₃, C₃ in 3D), with symmetry protection determining whether the invariant is Z₂ or Z. | MATH: Z₂ invariant = (−1)^ν where ν = ∫_BZ (1/2π) Tr[F] mod 2 (Chern parity); root system Weyl groups: A₂ (hexagonal, 6 roots), B₂ (square, 8 roots), G₂ (triangular, 12 roots); 3D: A₃ (fcc, 24 roots), B₃ (bcc, 18 roots), C₃ (simple cubic, 18 roots). Berry phase γ = ∮ A·dk mod 2π; for Z₂: γ = π (nontrivial) vs 0 (trivial). | CONNECTION: **Direct geometric harmony** — 2D root systems correspond to the three crystallographic lattices with rotational symmetries 6-fold (A₂), 4-fold (B₂), 3-fold (G₂). The Z₂ invariant's parity structure (0 or 1) mirrors the binary nature of root system Cartan matrices (off-diagonal entries 0, −1, −2). The 3+1D Z₂ toric code gravitational anomaly involves fermionic quasiparticles — linking to the 16-fold periodicity of K3 surfaces (Alessandra Sar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com