Brane Worlds and Strings in the Sector Calculus: Orbifolds, Orientifolds and the Sector Rules of Möbius–Klein Worldsheets
Year: 2026
Type: preprint
Abstract: The Möbius Klein Transformation (MKT) splits fields on a cover into the eigenspaces of a deck operation of order 2. The MKT untwisted sector has the deck character +1, the MKT twisted sector −1; for a deck translation the latter is the antiperiodic sector. For a reflection with fixed points the cover is an orbifold cover. This unifying review applies the split to brane worlds and strings. Every physics statement is known and attributed; 3 exact forms that follow in 1 line from cited results, and that we did not find stated in this form, are marked in place. The own content is the MKT reading, precise conditions and script checks. On S^1/Z_2 and S^1/(Z_2×Z_2') the 2 MKT characters fix the mode ladders, and their product is the Scherk–Schwarz twist. A single MKT character product rule gives the brane and trilinear couplings. It is necessary, not sufficient, and in a flat bulk it sees only the residual Z_2 of the Kaluza–Klein number. Kaluza–Klein parity is the MKT character of the half-period translation. The 2 chiralities of a bulk fermion are the 2 characters of the reflection, while the translation character is chirality-blind. In an orbifold GUT the X and Y bosons form the MKT twisted sector. The MKT twisted sum on a circle gives the massless Casimir ratio 2^(−d)−1, and the endpoint comparison of a Wilson-line potential is an MKT sign threshold. In gauge–Higgs unification the Higgs is the A_5 component on the broken generators: its odd index and odd group parity give the MKT character +1, the zero-mode condition. The Wilson line is a continuous MKT twist. The MKT sector label of a field on which −I acts as −1, such as the doublet, is not gauge invariant. On the warped orbifold the even and odd spin-2 towers interlace strictly. The (+,−) tower is the MKT twisted sector of the translation by 2πR, the composition of the 2 brane reflections, but not a uniform half shift. Under a Scherk–Schwarz twist ω=1/2 the graviton is MKT untwisted and the gravitino MKT twisted. A Casimir minimum of the radion needs a 2nd term with another radius dependence and the opposite sign, a 2nd sector sum or the bulk constant; without the bulk constant only the former works. The Euler characteristic, a Gauss–Bonnet integral, weights the string loops, fixes the T-duality factor of the dilaton and doubles on a cover. In a string orbifold the time twist is the MKT character; the string twisted sector has no field-theory counterpart. On the Scherk–Schwarz string the momentum and winding shifts are 2 MKT parities, exchanged by T-duality. The GSO sign of the twisted circle is a product of 2 MKT characters, while the Arf sign of a spin structure is a quadratic function of them, not a character. On orientable surfaces the Arf sign detects only the subgroup ,4of Z_8. On unoriented world sheets the full Pin− invariant is seen neither by the MKT characters nor by the orientation double cover. Every 1-loop tadpole contains both MKT character projections and string twisted sectors. For orientifolds the Klein bottle and Möbius amplitudes are MKT-weighted combinations of the 2 sectors of the worldsheet parity, the deck of the orientation double cover. The tadpole condition is an identity of such combinations. For D-branes the MKT sectors of the orientifold deck are the even and odd combinations of a brane-brane string and its image-image partner, and the type I′ interval is an orbifold. The warped hierarchy comes from curvature and brane distance, not from a sector rule, and the MKT adds no dynamics.
Author: Mark, Laszlo
Language: English
Subfield: Nuclear and High Energy Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225463163