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Background : In mathematical physics, 'anyons' in (2+1) dimensional systems are described by braided tensor categories. The anyon types correspond to the irreducible objects of the category. From such a category, one may derive the fusion ring, which is the ring generated by isomorphism classes of simple objects with addition corresponding to the direct sum, and multiplication corresponding to the tensor product. The tensor structure moreover induces some cohomological data on the fusion ring in the form of 'F-symbols' (also called 6j-symbols). The braiding induces additional data on the fusion ring in the form of 'R-symbols'.

In the physics literature, anyon theories are usually described by the fusion ring, F-symbols, and R-symbols alone. Let's call such data a `braided fusion ring'.

Question : If two braided tensor categories have gauge equivalent braided fusion rings, are they braided monoidally equivalent?

(The statement of the question was weakened in response to Danos' comment, see also point 2 below.)

Related facts :

  1. It is true that two semisimple tensor categories with gauge equivalent fusion rings and F-symbols are monoidally equivalent, see Proposition 1.1 of this paper.

  2. As pointed out by Danos in the comments, not all fusion rings actually come from a tensor category. Danos provides an explicit class of counter examples taken from this paper. This prompts an additional question : Are there braided fusion systems that do not come from a braided tensor category?

  3. The question is answered in the affirmative by Proposition 7.5.2. in the book Quantum Groups, Quantum Categories and Quantum Field Theory in the case where the fusion ring is isomorphic to $\mathbb{Z}(G)$ for a finite abelian group $G$.

  4. One may ask a similar question for modular tensor categories (cf. MathOverflow question) (in the physics literature, it is usually taken for granted that anyons are described by modular tensor categories). In that case the fusion ring comes with `modular data' in the form of S and T-matrices which seem to encode all physically relevant data about the anyons. However, there are inequivalent modular tensor categories with the same S and T-matrices, see this paper.

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  • $\begingroup$ Minor typo: In point 3) it should be "question for modular tensor categories" and again in the parenthesis. $\endgroup$
    – Danos
    Commented Dec 17, 2023 at 15:10
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    $\begingroup$ Also a comment on point 1) if I understand it correctly. The result shows uniqueness but not existence of the tensor category. For instance, given a fusion ring K (with F-matrices), it is not guaranteed that there exists a fusion category $\mathcal{C}$ which categorifies it. An example already appears in rank 2 (two simples) arxiv.org/abs/math/0203255. Namely, one can consider a fusion ring where $X^2 = 1 + n X$ but a fusion category with such a fusion ring only exists for $n=0,1$. $\endgroup$
    – Danos
    Commented Dec 17, 2023 at 16:21
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    $\begingroup$ The explicit arithmetic data that determines a modular fusion category is worked out in arxiv.org/abs/1305.2229. $\endgroup$
    – Rajath Radhakrishnan
    Commented Feb 11 at 14:16
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    $\begingroup$ If you have the fusion rules (the ring) and the associators (6j-symbols) satisfying the pentagon equation you can construct by hand a fusion category. If in addition you are given the "R-symbols" (i.e. braidings of simple objects) satisfying the appropriate hexagon equations, you can construct by hand a braiding on the fusion category. The problem considered in the paper cited by @Danos is about finding out whether, given just the ring, one can find compatible F-symbols. $\endgroup$
    – Manuel Araújo
    Commented Jul 8 at 7:53
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    $\begingroup$ Small correction to my comment: the 6j-symbols must satisfy the pentagon equations and also a simple condition that guarantees the resulting semisimple monoidal category will have duals. $\endgroup$
    – Manuel Araújo
    Commented Jul 8 at 8:20

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