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In 2D and 3D quibit models, string-net condensation can happen. In 3D or higher models, is it possible for surfaces (instead of just strings) to condense?

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Yes it seems that it is possible. See for example arXiv:cond-mat/0411752 where some models are constructed, but I (naively) think one can readily generalize the whole logic of Levin-Wen models to any dimension and any types of branes (the mathematical input might then be something more exotic than tensor categories?). But I think that excitations of these models will generically be extended objects, for example the boundaries of open membranes will be string-like object. One could also imagine that low-energy effective theories if these models might not be conventional (topological) field theories, but string- and brane-field theories (this is however not the case in the above reference it seems).

I don't know enough about this to say much more, but I know others on this site know a lot more. I hope they will give their take on this question.

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    $\begingroup$ In 3D, it is almost impossible to have a state with only string condensation but no membrane condensation, or only membrane condensation but no string condensation. In 4D space, it is possible to have a state with only membrane condensation but no string condensation. $\endgroup$
    – Xiao-Gang Wen
    Commented Dec 20, 2012 at 16:30

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