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

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.

$a\times b=c$ in $2+1$D TQFT

“…In the next section, we start with discrete gauge theories and explain how intuition in the theory of finite groups leads us to various answers to the above questions. Along the way, we prove various theorems about discrete gauge theories and fusion rules of the form (1.2) and (1.5) generalizing our work in [18]. Moreover, we discuss the role that subcategories and symmetries of discrete gauge theories play in such fusion rules.…”

“…In this section, we briefly review how to construct a discrete gauge theory given a finite gauge group, G (for a more succinct version of the review below, see [18]). Although much of what we say is a consequence of [13], we will follow the perspective in [17].…”

“…This triviality means that η g (h, k) can be expressed in terms of a one co-chain as follows: η g (h, k) = ǫg(h)ǫg(k) ǫg(h•k) . 18 2.2. Non-abelian Wilson lines and a × b = c…”

“…Since, by Cayley's theorem, every finite group is isomorphic to a subgroup of the symmetric group, S N , (for some N) it is natural to start our discussion with S N . In particular, 18 Note that 1 ǫ g is the irreducible projective representation of N g whose character is proportional to the trivial representation of N g . 19 In fact, we have C W ≃ Rep(G), where Rep(G) is the category of finite dimensional representations of G over C. 20 The Wilson lines braid non-trivially with other anyons in the theory (more formally: the Wilson line subcategory is Lagrangian and so the Müger center of C W is C W itself).…”

“…where S is the modular S-matrix? This is true in the context of discrete gauge theories with a simple gauge group [18]. However, non-trivial braiding does arise naturally in the context of the fusion of non-abelian electrically charged lines with non-abelian magnetically charged lines.…”