Indistinguishable Chargeon-Fluxion Pairs in the Quantum Double of Finite Groups

TL;DR

This paper investigates auto-equivalences in the quantum double of finite groups, revealing conditions under which chargeon-fluxion pairs become indistinguishable, especially in groups like S_3 and certain near-fields.

Contribution

It characterizes when chargeon-fluxion pairs are indistinguishable in the quantum double of specific finite groups, linking group structure to auto-equivalence existence.

Findings

Existence of auto-equivalences for groups like the semidirect product of finite field groups.

Chargeon-fluxion pairs in S_3 are indistinguishable.

Modular invariants correspond to groups isomorphic to certain near-fields.

Abstract

We consider the category of finite dimensional representations of the quantum double of a finite group as a modular tensor category. We study auto-equivalences of this category whose induced permutations on the set of simple objects (particles) are of the special form of PJ, where J sends every particle to its charge conjugation and P is a transposition of a chargeon-fluxion pair. We prove that if the underlying group is the semidirect product of the additive and multiplicative groups of a finite field, then such an auto-equivalence exists. In particular, we show that for S_3 (the permutation group over three letters) there is a chargeon and a fluxion which are not distinguishable. Conversely, by considering such permutations as modular invariants, we show that a transposition of a chargeon-fluxion pair forms a modular invariant if and only if the corresponding group is isomorphic to the semidirect product of the additive and multiplicative groups of a finite near-field.