Permutation orbifolds

TL;DR

The work develops a general theory of permutation orbifolds for arbitrary twist groups, delivering explicit formulas for primary counting, genus-one data, modular transformations, and fusion rules in terms of a unifying $\Lambda$-matrix and twisted-dimension framework. It clarifies how the orbifold central charge scales as $nc$ and uses covering-surface geometry to express genus-one characters and partition functions, while establishing Verlinde-type fusion through twisted dimensions. The paper also formulates tight arithmetic constraints on allowable twisted-dimension values and connects these to topological objects via Seifert-manifold partition functions. An explicit nonabelian $S_{3}$ example illustrates the concrete specialization of the general theory, including the enumeration of primaries and their genus-one data. Overall, the results provide a systematic toolkit for analyzing permutation orbifolds in RCFT and related 3D topological structures.

Abstract

A general theory of permutation orbifolds is developed for arbitrary twist groups. Explicit expressions for the number of primaries, the partition function, the genus one characters, the matrix elements of modular transformations and for fusion rule coefficients are presented, together with the relevant mathematical concepts, such as Lambda-matrices and twisted dimensions. The arithmetic restrictions implied by the theory for the allowed modular representations in CFT are discussed. The simplest nonabelian example with twist group S_3 is described to illustrate the general theory.