Symplectic Pfaffians and Z₂-Graded Topological Invariants in Polaritonic Graphene — E8 Intelligence Research

FINDING: Symplectic group Sp(2N) and its Z₂-graded Pfaffian structure underpin topological invariants in polaritonic graphene and symplectic category decompositions. | MATH: Sp(2N) = {M ∈ GL(2N, ℂ) | MᵀΩM = Ω}, Ω = [[0, I_N], [-I_N, 0]]; Pfaffian Pf(A)² = det(A) for skew-symmetric A; Z₂-grading: Pf(A) → ±Pf(A) under basis change with det = ±1; topological invariant ν = (1/2π)∮ dk · Tr[A(k)] mod 2, where A(k) is Berry connection in symplectic basis; Lagrangian submanifolds L ⊂ (M, ω) with dim L = N, ω|_L = 0. | CONNECTION: Sp(2N) contains maximal torus U(1)^N — roots are ±(e_i − e_j), ±(e_i + e_j) — a D_N root system (crystallographic, even lattice). The Z₂ grading maps to the parity of the Pfaffian, which for N=1 gives Sp(2) ≅ SU(2) with the golden ratio appearing in the SU(2) Clebsch-Gordan coefficients (e.g., 1/√2, √3/2 — but notably the ratio 0.618 appears in the SU(2) Wigner 6j symbols for j=1/2, j=1 recoupling). The polaritonic graphene honeycomb lattice has hexagonal (p6m) crysta Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com