Non-Abelian geometry and globally inequivalent symmetry reductions of the cylindrical Dirac doublet

• URL: http://arxiv.org/abs/2608.14041v1
• Date: Fri, 14 Aug 2026 07:38:07 GMT
• Title: Non-Abelian geometry and globally inequivalent symmetry reductions of the cylindrical Dirac doublet
• Abstract summary: Different exact cylindrical Dirac constructions are locally related within the same two-dimensional positive-energy sector.<n>We show that this equivalence can fail globally.<n>A single Dirac doublet admits symmetry resolutions that are locally equivalent but globally inequivalent.
• Score: 2.1686432495639045
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Different exact cylindrical Dirac constructions are locally related within the same two-dimensional positive-energy sector, suggesting that they might be merely alternative choices of basis. We show that this equivalence can fail globally. Treating the cylindrical Dirac doublet as a rank-two bundle over momentum space, we identify transverse helicity, a mass-dressed transverse integral, and helicity as distinct symmetry-selected rank-one reductions, whose normalized restrictions organize the internal doublet through a Pauli algebra. The positive-energy Dirac $SU(2)$ connection Abelianizes exactly on fixed-azimuth meridians in the transverse-helicity basis, while its full three-dimensional curvature remains genuinely non-Abelian for nonzero mass. We derive the corresponding azimuthal Wilson-loop spectrum in closed form. The three reductions then display sharply different global structures: the transverse-helicity splitting terminates on the momentum axis, the mass-dressed splitting extends smoothly and is Chern trivial for $m>0$, whereas helicity defines line bundles with opposite unit Chern numbers. Thus a single Dirac doublet admits symmetry resolutions that are locally equivalent but globally inequivalent.

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