
## 3. Braidings

1. ### Property F conjecture

If a fusion category has property F, then can’t use braiding alone for a universal quantum computer.

#### Problem 3.1.

[Rowell] Given a braided, weakly integral fusion category, is the image of the braid group finite?

Is braided and weakly integral fusion equivalent to finite image of the braid group?
• #### Problem 3.2.

Is there a physical model which gives infinite image for the braid group?
• #### Problem 3.3.

[Hagge, Wang] For unitary theories, we can choose a gauge so the $F$ matrices formed by the $6j$ symbols are unitary, and the braiding matrices are diagonal with respect to a certain basis. Can this happen for some non-unitary fusion category?

Is the unitarity of the $F$ matrices equivalent to unitarity?
Snyder: if $F$ matrices are unitary, and dimensions are positive, and maybe something about $\theta$’s, then it is unitary.

Cite this as: AimPL: Classifying fusion categories, available at http://aimpl.org/fusioncat.