Non-Abelian Anyons and Some Cousins of the Arad-Herzog Conjecture

TL;DR

This paper explores the implications of the Arad-Herzog conjecture for non-abelian anyons in 2+1D gauge theories, proving related statements and providing physical intuition, linking group theory conjectures with quantum physics phenomena.

Contribution

It establishes new connections between the AH conjecture and non-abelian anyons, proving related statements and offering physical insights into their validity.

Findings

Proved related statements about finite simple groups and gauge theories.

Provided physical intuition supporting the AH conjecture.

Linked lack of certain dualities to the conjecture's validity.

Abstract

Long ago, Arad and Herzog (AH) conjectured that, in finite simple groups, the product of two conjugacy classes of length greater than one is never a single conjugacy class. We discuss implications of this conjecture for non-abelian anyons in 2+1-dimensional discrete gauge theories. Thinking in this way also suggests closely related statements about finite simple groups and their associated discrete gauge theories. We prove these statements and provide some physical intuition for their validity. Finally, we explain that the lack of certain dualities in theories with non-abelian finite simple gauge groups provides a non-trivial check of the AH conjecture.