Local Chirality from Defects in an E8 Framework: Jackiw-Rebbi Zero Mode and Conditional Sector Selection

We study a topological route to chiral fermions in a framework with a parent E8 symmetry. The positive result is deliberately bounded: in the Jackiw–Rebbi 1+1D analogue, a topological defect localizes exactly one normalizable fermionic zero mode, with chirality fixed by the sign of the defect charge. The branching 248 → (78,1)+(1,8)+(27,3)+(27bar,3bar) preserves a natural route to three families if the light sector is (27,3), but the net selection of that sector over its conjugate is not derived: topology alone does not bias the net sign, symmetric Dirac mixing lifts both chiralities together (argued from the singular values), and the mechanisms that do select a sector require a CP/theta ingredient or a mass/twist/condensate spurion not currently derived. The honest status is: local chirality from defects is derived in an analogue; net sector chirality is conditional; observable electroweak chirality remains open. Version 3 — correction notice (physics audit). (1) The breaking E8 → E6 × SU(3)_F has NO codimension-two defects: the unbroken subgroup is H = (E6×SU(3))/Z3, so π1(E8/H) = 0 — strings do not exist — while π2(E8/H) = Z3: what exists are Z3 monopoles (codimension three). This corrects the earlier internal statement that the coset was topologically empty, and it is the only correction of this audit that OPENS a route (whether the monopole sector carries chiral zero modes is an open question). (2) Dimension counting: a codimension-two defect in D = 4 carries 1+1-dimensional, not 4D, chiral modes; 4D chirality on a defect needs codim-1 in D = 5 (Kaplan) or codim-2 in D = 6 (Callan–Harvey). (3) "One zero mode per unit charge" is withdrawn: for φ_Q = Qv·tanh(mx) the charge is an amplitude, not a winding number, and there is exactly one normalizable mode for every Q ≠ 0 (verified by diagonalizing the SUSY-QM partner Hamiltonians). (4) The asymmetric mirror mass is gauge-forbidden, not merely underived: 27⊗27 = 27bar+351+351' contains no E6 singlet. (5) Basis-invariance: only singular values of a Dirac mass matrix are physical. (6) The Yukawa 27_F·27_F·27_H vanishes for an SU(3)_F-singlet Higgs; 27_H must sit in the 6. (7) k_min = 1, not 3: π3(E8) = Z with trivial centre. (8) "Fermions only in real representations of E8" is vacuous (every E8 representation is real). Plus asymptotics, reference and compilation fixes.

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-08-24
DOI
https://doi.org/10.5281/zenodo.20075672
Primary Topic
Quantum Chromodynamics and Particle Interactions
Type
preprint
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preprint

Local Chirality from Defects in an E8 Framework: Jackiw-Rebbi Zero Mode and Conditional Sector Selection

O. F. C.
Zenodo (CERN European Organization for Nuclear Research)
Quantum Chromodynamics and Particle Interactions
preprint

Local Chirality from Defects in an E8 Framework: Jackiw-Rebbi Zero Mode and Conditional Sector Selection

O. F. C.
preprint en

Abstract

We study a topological route to chiral fermions in a framework with a parent E8 symmetry. The positive result is deliberately bounded: in the Jackiw–Rebbi 1+1D analogue, a topological defect localizes exactly one normalizable fermionic zero mode, with chirality fixed by the sign of the defect charge. The branching 248 → (78,1)+(1,8)+(27,3)+(27bar,3bar) preserves a natural route to three families if the light sector is (27,3), but the net selection of that sector over its conjugate is not derived: topology alone does not bias the net sign, symmetric Dirac mixing lifts both chiralities together (argued from the singular values), and the mechanisms that do select a sector require a CP/theta ingredient or a mass/twist/condensate spurion not currently derived. The honest status is: local chirality from defects is derived in an analogue; net sector chirality is conditional; observable electroweak chirality remains open. Version 3 — correction notice (physics audit). (1) The breaking E8 → E6 × SU(3)_F has NO codimension-two defects: the unbroken subgroup is H = (E6×SU(3))/Z3, so π1(E8/H) = 0 — strings do not exist — while π2(E8/H) = Z3: what exists are Z3 monopoles (codimension three). This corrects the earlier internal statement that the coset was topologically empty, and it is the only correction of this audit that OPENS a route (whether the monopole sector carries chiral zero modes is an open question). (2) Dimension counting: a codimension-two defect in D = 4 carries 1+1-dimensional, not 4D, chiral modes; 4D chirality on a defect needs codim-1 in D = 5 (Kaplan) or codim-2 in D = 6 (Callan–Harvey). (3) "One zero mode per unit charge" is withdrawn: for φ_Q = Qv·tanh(mx) the charge is an amplitude, not a winding number, and there is exactly one normalizable mode for every Q ≠ 0 (verified by diagonalizing the SUSY-QM partner Hamiltonians). (4) The asymmetric mirror mass is gauge-forbidden, not merely underived: 27⊗27 = 27bar+351+351' contains no E6 singlet. (5) Basis-invariance: only singular values of a Dirac mass matrix are physical. (6) The Yukawa 27_F·27_F·27_H vanishes for an SU(3)_F-singlet Higgs; 27_H must sit in the 6. (7) k_min = 1, not 3: π3(E8) = Z with trivial centre. (8) "Fermions only in real representations of E8" is vacuous (every E8 representation is real). Plus asymptotics, reference and compilation fixes.

Zenodo (CERN European Organization for Nuclear Research)
Quantum Chromodynamics and Particle Interactions
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