Permutation orbifolds

TL;DR

This paper develops a comprehensive mathematical framework for permutation orbifolds in conformal field theory, providing explicit formulas and exploring modular representation restrictions, with an example involving the symmetric group S_3.

Contribution

It introduces a general theory for permutation orbifolds with arbitrary twist groups, including explicit formulas and mathematical tools for analyzing their properties.

Findings

Derived formulas for primaries, partition functions, and modular transformations.

Explored arithmetic restrictions on modular representations in CFT.

Provided an example with the nonabelian group S_3 to illustrate the theory.

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.