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

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Z₂ topological invariants in 2D and 3D band structures map directly onto root-system classifications, with symmetry protection determining whether the invariant is Z₂ or Z. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22235753

September 1, 2026Open Access

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

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Key Points

To investigate whether Z₂ topological invariants in two- and three-dimensional electronic band structures correspond to Lie algebra root-system classifications.
Mathematical analysis linking Berry phase and Chern parity formulation of Z₂ invariants to root-system classifications across lattice symmetries.
Evaluated crystallographic geometries including 2D systems (A₂, B₂, G₂) and 3D systems (A₃, B₃, C₃) against root system Weyl groups and Cartan matrices.
Demonstrated that 2D Z₂ invariants directly correspond to hexagonal (A₂), square (B₂), and triangular (G₂) crystallographic lattices governed by rotational symmetry.
Identified that 3D invariants map directly to face-centered cubic (A₃), body-centered cubic (B₃), and simple cubic (C₃) root structures.
Showed that the binary parity structure of Z₂ topological invariants mirrors the discrete off-diagonal entries (0, −1, −2) of root system Cartan matrices.

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Theoretical analysis demonstrates mapping of topological invariants to root systems in crystal lattices, revealing geometric symmetry links.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a993592636c6408cfa7df27 https://doi.org/10.5281/zenodo.22235753

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

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

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