In mathematics and physics, an anyon is a type of projective representation of a Lie group.
In detail, there are projective representations of SO(2,1) which don't arise from linear representations of SO(2,1), or of its double cover, Spin(2,1). These representations are called anyons.
The topological reason behind the phenomenon is this: the first homotopy group of SO(2,1) (and also Poincaré(2,1)) is Z (infinite cyclic). This means that Spin(2,1) is not the universal cover: it is not simply connected. On the other hand, for n > 2, for SO(n,1) and Poincaré(n,1), it's only Z2 (cyclic of order 2); meaning that the spin group is simply connected.
This mathematical concept becomes useful in the physics of two-dimensional systems such as sheets of graphite or the quantum Hall effect. In space of three dimensions (or more), elementary particles have tightly constrained quantum numbers and, in particular, are restricted to being fermions or bosons. In two-dimensional systems, however, quasiparticles are observed whose quantum states range continuously between fermionic and bosonic, taking on any quantum value in between. Frank Wilczek coined the term "anyons" in 1982 to describe such particles.
External links
Interview with Frank Wilczek on anyons and superconductivity
See also: plekton.