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\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

3. Braidings

    1. Property F conjecture

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

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      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?


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          Problem 3.2.

          Is there a physical model which gives infinite image for the braid group?


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              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.