The Non-Abelian Möbius Klein Transformation: Sectors as Irreducible Representations and Fusion as Selection

The Möbius Klein Transformation (MKT) sorts the fields on a non-orientable space by the deck group of its orientable double cover. That group is cyclic of order 2, and every rule of the MKT canon is abelian: a sector is a character, and a selection rule is a product of signs. This article lets the deck group be any finite group, on free covers and on covers branched along points, knots and graphs. A sector is then an isotypic component, a field in it is a multiplet, and the product of signs becomes the fusion rule of the irreducible representations. On graph covers with the symmetric group of 3 letters the spectrum splits into the trivial, the sign and twice the standard sector, and the 3-sheeted cover keeps the trivial and the standard one. The cover sum of 3 sector fields vanishes whenever their tensor product misses the trivial representation, and generically only then; a bookkeeping by determinant characters gets this wrong. Sunada's isospectral pairs are covers with equal sector content, and the incidence matrix of the Fano plane is the transplantation. Over a smooth base the exponents of a sector at a branch edge follow the eigenvalues of the meridian, so a reflection gives a square-root exponent. At a vertex with a polyhedral local group they are the degrees of the spherical harmonics that contain the representation. Dihedral covers of the trefoil and the figure-eight knot, built from Fox colourings, split into the base and its 2-dimensional sectors. On a non-orientable flat 4-manifold with dihedral holonomy of order 8 the orientation character and the holonomy act together on every tensor slice. Around 2 quaternion defects frame vectors see only characters of the base, and only a spinor field sees the non-abelian sector.