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


By [Ostrik's work](https://arxiv.org/abs/math/0111139), we know these module categories correspond to separable algebras in **Fib**. I do not believe such things have been classified.


[Davydov and Booker](https://arxiv.org/abs/1103.3537) 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**.



Module categories for Fibonacci anyons
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. 
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> 
> 


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.


