The torus and the Klein Bottle amplitude of permutation orbifolds

TL;DR

This paper derives formulas for calculating the torus and Klein bottle amplitudes in permutation orbifolds of RCFTs, linking them to properties of the original theory and the twist group, with explicit examples for Z2 orbifolds.

Contribution

It provides explicit formulas for amplitudes and counts of self conjugate primaries in permutation orbifolds, extending understanding of their structure.

Findings

Derived formulas for torus and Klein bottle amplitudes in permutation orbifolds.

Expressed the number of self conjugate primaries as a polynomial in original theory parameters.

Illustrated results with explicit Z2 orbifold examples.

Abstract

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 $Z_2$ orbifold illustrate the general results.