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"Indistinguishable chargeon-fluxion pairs in the quantum double of finite groups", Salman Beigi, Peter Shor, Daniel Whalen, 2010

Reviewed December 31, 2023

Citation: Beigi, Salman, Peter W. Shor, and Daniel Whalen. "Indistinguishable chargeon-fluxion pairs in the quantum double of finite groups." arXiv preprint arXiv:1002.4930 (2010).

Web: https://arxiv.org/abs/1002.4930

Tags: SPT/SETs, Modular-tensor-categories,

This paper gives a a nice construction of form charge-flux duality which applies to a certain class of modular tensor categories. Namely, it takes in a group G with a special form and constructs an auto-equivalence of Z(Vec_G) such that an electric charge (chargeon) is interchanged with a magnetic charge (fluxion). Here, a chargeon refers to an element of Z(Vec_G) whose underlying object is trivial and fluxion refers to any object whose underlying object is non-trivial.

To be precise, the induced map on objects is a composition PJ where J sends every particle to its charge-conjugate (i.e. inverting the braiding isomorphisms) and P is the transposition of a specific chargeon with a specific fluxion. A key part of this paper comes from explicitly constructing these distinguished chargeons/fluxions, which is a non-trivial matter.

This charge/fux duality exists when G is isomorphic to a distinguished semidirect product Fq^+ \rtimes Fq^x where Fq is a finite field. The cocycle in this semidirect product is given by multiplication. This charge/flux duality cannot be generalized to arbitrary groups, as shown in the paper.

One very interesting aspect of this paper is that it gives a connection between Z2 and S3. When q=2, Fq^+ \rtimes Fq^x is Z2. The charge/flux duality here is the standard e/m duality. When q=3, one obtains the group S3 and the charge/flux duality is more non-trivial. This is one case in which the topological order associated to S3 is nicer than the topological order associated to D4!