  Fractional statistics and quantum chaos are both phenomena associated with
the non-local storage of quantum information. In this article, we point out a
connection between the butterfly effect in (1+1)-dimensional rational conformal
field theories and fractional statistics in (2+1)-dimensional topologically
ordered states. This connection comes from the characterization of the
butterfly effect by the out-of-time-order-correlator proposed recently. We show
that the late-time behavior of such correlators is determined by universal
properties of the rational conformal field theory such as the modular S-matrix
and conformal spins. Using the bulk-boundary correspondence between rational
conformal field theories and (2+1)-dimensional topologically ordered states, we
show that the late time behavior of out-of-time-order-correlators is
intrinsically connected with fractional statistics in the topological order. We
also propose a quantitative measure of chaos in a rational conformal field
theory, which turns out to be determined by the topological entanglement
entropy of the corresponding topological order.
