The torus and the Klein bottle amplitude of permutation orbifolds

The torus and the Klein bottle amplitude coefficients are computed in permutation orbifolds of RCFT-s in terms of the same quantities in the original theory and the twist group. An explicit expression is presented for the number of self conjugate primaries in the orbifold as a polynomial of the total number of primaries and the number of self conjugate ones in the parent theory. The formulae in the Z2 orbifold illustrate the general results. Permutation orbifolds have been investigated in the last couple of years, as they are not only a special class of Rational Conformal Field Theories, but they are also closely related to second quantisation of strings ([1]). The one loop amplitude is a starting point there, its form that is, its dependence on the characters of the primary fields of the corresponding Conformal Field Theory is determined from general principles. Finding the explicit dependence amounts to writing down the coefficients of the corresponding linear combination of the characters in the open, the sesquilinear combination in the closed case, respectively. This is possible in a permutation orbifold and the topic of this paper in terms of the same coefficients of the ”ascendant” CFT. It is useful for getting information about the structure of orbifolds and for providing explicit formulae which can be subject for resting conjectures about the further structure of the amplitude coefficients. For any RCFT C and any permutation group Ω < Sn a new CFT C ≀ Ω can be constructed by taking the n-fold tensor product of C and identifying states according to the orbits of the standard action of Ω. The new theory is called the permutation orbifold of C ([2], [3]) and every relevant quantity (e.g. conformal weights, genus one characters of the primaries, the matrix elements of the modular transformations, the partition function etc.) is completely determined in terms of the corresponding quantities of C and the twist group Ω. The general case (when the twist group is nonabelian) was discussed in [4].