Rowell–Wang localization conjecture

Determine whether, for every simple object X in a unitary braided fusion category, the sequence of braid-group representations on the endomorphism algebras of tensor powers of X is localizable by a unitary Yang–Baxter operator if and only if the square of the Frobenius–Perron dimension of X is an integer.

Background

The localization problem asks when braid representations arising from anyon models or unitary braided fusion categories can be realized by local operators acting on tensor powers of a fixed Hilbert space. The proposed criterion relates this operator-theoretic property to the categorical invariant FP-dimension.

The paper verifies particular localization cases, including a qutrit BMW example and a quaternionic Family III example, but these examples do not settle the criterion for arbitrary simple objects in arbitrary unitary braided fusion categories.