This is essentially an addition to the list of @4tnemele

I'd like to add some earlier work to this list, namely Discrete Gauge Theory.

Discrete gauge theory in 2+1 dimensions arises by breaking a gauge symmetry with gauge group $G$ to some lower **discrete** subgroup $H$, via a Higgs mechanism. The force carriers ('photons') become massive which makes the gauge force ultra-short ranged. However, as the gauge group is not completely broken we still have the the Aharanov-Bohm effect. If H is Abelian this AB effect is essentially a 'topological force'. It gives rise to a phase change when one particle loops around another particle. This is the idea of fractional statistics of Abelian anyons.

The particle types that we can construct in such a theory (i.e. the one that are "color neutral") are completely determined by the residual, discrete gauge group $H$. To be more precise: a particle is said to be charged if it carries **a representation of the group H**. The number of different particle types that carry a charge is then equal to the number of irreducible representations of the group H. This is similar to ordinary Yang-Mills theory where charged particles (quarks) carry the fundamental representation of the gauge group (SU(3). In a discrete gauge theory we can label all possible charged particle types using the representation theory of the discrete gauge group H.

But there are also other types of particles that can exist, namely those that carry flux. These flux carrying particles are also known as magnetic monopoles. In a discrete gauge theory the flux-carrying particles are labeled by the **conjugacy classes of the group H**. Why conjugacy classes? Well, we can label flux-carrying particles by elements of the group H. A gauge transformation is performed through conjugacy, where $ |g_i\rangle \rightarrow |hg_ih^{-1}\rangle $ for all particle states $|g_i\rangle$ (suppressing the coordinate label). Since states related by gauge transformations are physically indistinguishable the only unique flux-carrying particles we have are labeled by conjugacy classes.

Is that all then? Nope. We can also have particles which carry both charge and flux -- these are known as dyons. They are labeled by both an irrep and a conjugacy class of the group $H$. But, for reasons which I wont go into, we cannot take all possible combinations of possible charges and fluxes.

(It has to do with the distinguishability of the particle types. Essentially, a dyon is labeled by $|\alpha, \Pi(g)\rangle$ where $\alpha$ is a conjugacy class and $\Pi(N(g))$ is a representation of the associated normalizer $N(\alpha)$ of the conjugacy class $\alpha$.)

The downside of this approach is the rather unequal setting of flux carrying particles (which are labeled by conjugacy classes), charged particles (labeled by representations) and dyons (flux+compatible charge). A unifying picture is provided by making use of the (quasitriangular) Hopf algebra $D(H)$ also known as a quantum double of the group $H$.

In this language *all* particles are (irreducible) representations of the Hopf algebra $D(H)$. A Hopf Algebra is endowed with certain structures which have very physical counterparts. For instance, the existence of a **tensor product** allows for the existence of **multiple particle configurations** (each particle labeled by their own representation of the Hopf algebra). The **co-multiplication** then defines **how the algebra acts** on this tensored space. the existence of an **antipode** (which is a certain mapping from the algebra to itself) ensures the existence of an **anti-particle**. The existence of a unit labels the vacuum (=trivial particle).

We can also go beyond the structure of a Hopf algebra and include the notion of an R-matrix. In fact, the quasitriangular Hopf Algebra (i.e. the quantum double) does precisely this: add the R-matrix mapping. This R-matrix describes what happens when one particle loops around another particle (braiding). For non-Abelian groups $H$ this leads to non-Abelian statistics. These quasitriangular Hopf algebras are also known as quantum groups.

Nowadays the language of discrete gauge theory has been replaced by more general structures, referred to by topological field theories, anyon models or even modular tensor categories. The subject is huge, very rich, very physical and a lot of fun (if you're a bit nerdy ;)).

Sources:

http://arxiv.org/abs/hep-th/9511201 (discrete gauge theory)

http://www.theory.caltech.edu/people/preskill/ph229/ (lecture notes: check out chapter 9. Quite accessible!)

http://arxiv.org/abs/quant-ph/9707021 (a simple lattice model with anyons. There are definitely more accessible review articles of this model out there though.)

http://arxiv.org/abs/0707.1889 (review article, which includes potential physical realizations)

This post imported from StackExchange Physics at 2015-11-01 19:23 (UTC), posted by SE-user Olaf