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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!