By Morita equivalent I mean that there is an invertible bi-module between the two fusion categories. [Feel free to replace the Drinfeld centers being "equal" by an appropriate categorial notion of "equivalent"; for me (braided) fusion categories are just F-tensors (and R-tensors), in which case it's really "equal" (up to a gauge on the fusion vector spaces)]

I'm asking because there is a very direct physical translation of that statement:

"Two gapped 2+1-dimensional non-chiral topologically ordered systems are in the same phase if their anyon content is the same."

By "in the same phase" here I mean equivalence under a short-range local unitary circuit, or in other words an invertible domain wall. The statement seems so accepted in the physics community that hardly anyone talks about it. Still I never heard a simple argument why it should be so.

For the Turaev-Viro-Levin-Wen fixedpoint models, the physical systems are constructed from fusion categories, and invertible domain walls can be constructed from invertible bi-modules.


In the non-unitary setting ENO proved that if $Z(C)$ and $Z(D)$ are equivalent as braided tensor categories, then C and D are Morita equivalent. This is Theorem 3.1 of this paper. Note that they say the result was already known to Kitaev and Müger. This is also an iff, though the other direction is easier.

The analogous result also holds in the unitary setting. That is if $Z(C)$ and $Z(D)$ are equivalent as unitary braided tensor categories then C and D are unitarily Morita equivalent. (Unitary Morita equivalence means that the actions on the bimodule are compatible with the star structure.) I don't know whether this has appeared in the literature, but the proof is completely analogous just replacing "algebra" with "$C^*$-algebra" (or Q-System if you prefer) everywhere. Again this is an iff.

You do need to be careful that you put in the word "unitary" everywhere. For example, it's not at all clear to me whether two unitary fusion categories being algebraically Morita equivalent is enough to say that they're unitarily Morita equivalent.

(Aside: I'm not sure at all what you mean by "really are equal" since they're really not equal, you still may need to gauge.)

  • $\begingroup$ jup, I should have said "equal up to a gauge on the fusion vector spaces", and I guess "unitary Morita equivalence" is actually the thing I'm interested in. Thanks! $\endgroup$
    – Andi Bauer
    May 4 '18 at 8:52

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