The Real Möbius Klein Transformation: Antilinear Gluing, Prym Splitting and Klein Surfaces

A twisted space can be glued to itself in 2 ways. A unitary gluing splits the functions on its double cover into an untwisted and a twisted sector, and a symmetric operator cannot connect the 2. An antilinear gluing, which conjugates the field as it reflects the space, splits nothing. It imposes a reality condition instead. This article develops the transform that belongs to the antilinear gluing. The functions it admits form a real vector space of half the real dimension of the free fields on the cover. A closed transform pair on this space has a real orthogonal basis, a Parseval identity and a fast algorithm with 2 complex transforms of half the cover length. An operator commutes or anticommutes with the antilinear gluing, or neither. Its class is the product of its unitary parity and its reality sign. Matrix elements between admissible states are then real or purely imaginary: reality replaces vanishing. When the gluing squares to minus one, every level is doubly degenerate. In complex dimension one the same gluing makes the Möbius strip and the Klein bottle into Klein surfaces, the quotients of an annulus and a torus by an antiholomorphic involution. A dianalytic function on them obeys the coefficient condition of the transform pair, and holomorphy is not a sector. On hyperelliptic double covers the holomorphic differentials split into the quotient part and the Prym part, and the halving of the cover appears there as it does on the lattice. A Dirichlet problem on the conformal Möbius strip is solved from data on its single boundary circle. A complex field with the antilinear gluing has the vacuum energy of 1 real field on the cover, the sum of its 2 unitary sector energies. Every statement is checked by a deterministic script with counter-checks.