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What are the module categories over the modular tensor category Fib of Fibonacci anyons?

By Ostrik's work, we know these module categories correspond to separable algebras in Fib. I do not believe such things have been classified.

Davydov and Booker show there are no nontrivial commutative separable algebras in Fib, but I do not see that they make a clear statement for this more general case, without commutativity.

My guess is that there are indeed no nontrivial module categories for Fib.

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There is only one equivalence class of indecomposable module categories, namely the trivial one.

Let us look into the possible algebras. They are $1$ and $1\oplus \tau$, and both have a unique algebra structure. For the first this is trivial and for the second is basically saying that the $A_4$ subfactor is unique. But $1\oplus\tau =\tau\otimes \bar\tau$ so it coincides with the internal action hom $\underline{\mathrm{Hom}}(\tau,\tau)$ of Fib as a module category over itself.

Here is another proof. The only possible connected étale (commutative) algebra is $1$, as the OP points out. Then by Corollary 3.8 in

Davydov, A.; Nikshych, D.; Ostrik, V.: On the structure of the Witt group of braided fusion categories. Selecta Math. (N.S.) 19 (2013), no. 1, 237–269.

module categories are in one-to-one correspondence with braided autoequivalences of $\mathrm{Fib}\to\mathrm{Fib}$ for which there exist only the trivial one.

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  • $\begingroup$ If you want to see more detail for this style of argument (eg, why are those the only possible algebras) we do a more elaborate example here. $\endgroup$ – Noah Snyder Jul 6 '16 at 19:43

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