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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2012

    brief entry on Turaev-Viro model, an entry long overdue. But for the moment it just records some references.

    Bruce Bartlett has a comment on what is currently the last of these references and he will post it below in a moment…

    • CommentRowNumber2.
    • CommentAuthorBruce Bartlett
    • CommentTimeOct 17th 2012
    Thanks to the elegant Kirillov and Kirillov-Balsam papers, given a spherical category A, we now know of four ways of setting up the 3d TQFT vector space of a surface Sigma:

    1. As the Reshetikhin-Turaev vector space, based on the modular category Z(A) : has basis given by loops labelled by the simple objects in Z(A) running around the holes in Sigma.

    2. As the TV vector space: triangulate Sigma, and sum over all tensor products of labellings of the edges by simple objects in A. There is a canonical projector on this space; the image of this projector is the TV vector space.

    3. As the Levin-Wen ground state space of a Hamiltonian operator operating on a triangulation of Sigma, based on the 6j symbols of A.

    4. As the Kitaev-Kirillov-Levin-Wen string-net space of linear combinations of string diagrams drawn on Sigma up to local moves of string diagrams.

    The string-net space (number 4) is the most intriguing to me.
    • CommentRowNumber3.
    • CommentAuthorBruce Bartlett
    • CommentTimeOct 17th 2012
    For instance, the string-net space seems to be an elegant alternative approach to the concept of "holonomy" of nonabelian gerbes. One might say that the holonomy -is- the string net space.
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 18th 2014

    have added more references, with brief comments, to the entry Turaev-Viro model

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