Chern-Simons Theory & Anyons

In 2+1D, particle statistics are not restricted to bosons/fermions. Anyons acquire phase e^{iθ} under exchange. Chern-Simons theory with level k gives θ=π/k. Watch worldlines braid and accumulate topological phase.

Accumulated phase
e^{iπ/2}
= 0.000 + i·1.000
Phase = n·θ = n·(π/m) for ν=1/m FQH
Chern-Simons action:
S = (k/4π)∫ A∧dA

Level k → θ = π/k
ν=1/m FQH: θ=π/m, q=e/m
Topological spin: h=θ/(2π)