The classification of $3^n$ subfactors and related fusion categories

We investigate a (potentially infinite) series of subfactors, called $3^n$ subfactors, including $A_4$, $A_7$, and the Haagerup subfactor as the first three members corresponding to $n=1,2,3$. Generalizing our previous work for odd $n$, we further develop a Cuntz algebra method to construct $3^n$ subfactors and show that the classification of the $3^n$ subfactors and related fusion categories is reduced to explicit polynomial equations under a mild assumption, which automatically holds for odd $n$. In particular, our method with $n=4$ gives a uniform construction of $4$ finite depth subfactors, up to dual, without intermediate subfactors of index $3+\sqrt{5}$. It also provides a key step for a new construction of the Asaeda–Haagerup subfactor due to Grossman, Snyder, and the author.