Topological Insulators and Cartan's Symmetric Spaces: A 10-Fold Isomorphism — E8 Intelligence Research

Abstract

FINDING: The Altland-Zirnbauer (AZ) 10-fold classification of topological insulators/superconductors is mathematically isomorphic to the 10 symmetric spaces of Cartan, with the periodic structure governed by Bott periodicity (period 8 for real, period 2 for complex classes). | MATH: The 10 AZ classes correspond to the 10 compact symmetric spaces: - Complex classes (A, AIII): period 2, via \( \mathbb{Z} \times BU \) and \( U/U\times U \) - Real classes (AI, BDI, D, DIII, AII, CII, C, CI): period 8, via real K-theory \( KO^{-n} \cong KO^{-n-8} \) Bott periodicity: \( \pi_n(O) \cong \pi_{n+8}(O) \), \( \pi_n(U) \cong \pi_{n+2}(U) \). The 10 spaces: \( U(n) \), \( U(p+q)/U(p)\times U(q) \), \( O(n) \), \( O(p+q)/O(p)\times O(q) \), \( Sp(n) \), \( Sp(p+q)/Sp(p)\times Sp(q) \), \( U(n)/O(n) \), \( U(2n)/Sp(n) \), \( U(n)/U(p)\times U(q) \), \( U(2n)/Sp(2n) \). | CONNECTION: The period-8 structure mirrors the octonionic (Cayley) division algebra — the only normed division algebras (R Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com