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

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The Altland-Zirnbauer (AZ) 10-fold classification of topological insulators and superconductors is mathematically isomorphic to the 10 symmetric spaces of Cartan, with its periodic structure governed by Bott periodicity. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22075868

August 24, 2026Open Access

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

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Key Points

Map the Altland-Zirnbauer 10-fold classification of topological insulators and superconductors directly to Cartan's compact symmetric spaces.
Analyzed the topological classification of quantum systems using topological K-theory and Bott periodicity.
Mapped real and complex Altland-Zirnbauer symmetry classes to classical Lie group symmetric coset spaces.
Demonstrated that the 10 Altland-Zirnbauer classes correspond directly to Cartan's 10 compact symmetric spaces.
Showed that topological periodicity is governed by period-8 Bott periodicity for real classes and period-2 for complex classes, mirroring normed division algebras.

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Implication

Theoretical analysis reveals a mathematical isomorphism between topological insulator classes and Cartan symmetric spaces, highlighting underlying geometric periodicity.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a8e9b58451774b83f3b4092 https://doi.org/10.5281/zenodo.22075868

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

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

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