Simple currents versus orbifolds with discrete torsion -- a complete classification

TL;DR

This paper provides a comprehensive classification of simple current modular invariants using orbifold techniques, establishing a correspondence with subgroups of the center with discrete torsion and proving conjectures about their enumeration.

Contribution

It extends previous classifications to arbitrary centers, offers explicit formulas, and clarifies the role of discrete torsion in simple current invariants.

Findings

Complete classification of simple current invariants for arbitrary centers.

Established a one-to-one correspondence with subgroups with discrete torsion.

Proved monodromy independence of the total number of invariants.

Abstract

We give a complete classification of all simple current modular invariants, extending previous results for $(\Zbf_p)^k$ to arbitrary centers. We obtain a simple explicit formula for the most general case. Using orbifold techniques to this end, we find a one-to-one correspondence between simple current invariants and subgroups of the center with discrete torsions. As a by-product, we prove the conjectured monodromy independence of the total number of such invariants. The orbifold approach works in a straightforward way for symmetries of odd order, but some modifications are required to deal with symmetries of even order. With these modifications the orbifold construction with discrete torsion is complete within the class of simple current invariants. Surprisingly, there are cases where discrete torsion is a necessity rather than a possibility.