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 (3) "One zero mode per unit charge" is withdrawn: for φ_Q = Qv·tanh(mx) the charge is an amplitude, not a winding number, and there is exactly one normalizable mode for every Q ≠ 0 (verified by diagonalizing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.
Authors
- O. F. C.
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