  Governed by locality, we explore a connection between unitary braid group
representations associated to a unitary $R$-matrix and to a simple object in a
unitary braided fusion category. Unitary $R$-matrices, namely unitary solutions
to the Yang-Baxter equation, afford explicitly local unitary representations of
braid groups. Inspired by topological quantum computation, we study whether or
not it is possible to reassemble the irreducible summands appearing in the
unitary braid group representations from a unitary braided fusion category with
possibly different positive multiplicities to get representations that are
uniformly equivalent to the ones from a unitary R-matrix. Such an equivalence
will be called a localization of the unitary braid group representations. We
show that the q=exp(\pi i/6) specialization of the unitary Jones representation
of the braid groups can be localized by a unitary $9\times 9$ R-matrix.
Actually this Jones representation is the first one in a family of theories
(SO(N),2) for an odd prime N>1, which are conjectured to be localizable. We
formulate several general conjectures and discuss possible connections to
physics and computer science.
