In 3+ spatial dimensions, particle exchange yields only fermions ($-1$) or bosons ($+1$): the configuration space of $N$ identical particles has fundamental group $S_N$.

In 2D, the configuration space has fundamental group the braid group $B_N$ — exchanging particles is a path in 2D that depends on whether the path goes clockwise or counterclockwise. Phases under exchange can be any $e^{i\theta}$ — these are abelian anyons. Non-abelian anyons transform by matrices, not phases, and underlie topological quantum computation.

Realizations: fractional quantum Hall states (Laughlin $\nu = 1/3$ quasi-particles have $\theta = \pi/3$); Majorana zero modes in topological superconductors are non-abelian (Ising anyons).

The exchange phase is a topological invariant — robust to local perturbations. This robustness is what makes anyonic systems candidates for fault-tolerant quantum computers: information is encoded non-locally in the topology of the worldlines.

Interactive: braiding two anyons in 2D

Drag the slider to braid particle 1 around particle 2. The accumulated phase = exchange angle × winding.