Classification of modular data up to rank 12

Abstract

We use the computer algebra system GAP to classify modular data up to rank 12. This extends the previously obtained classification of modular data up to rank 6. Our classification includes all the modular data from modular tensor categories up to rank 12, with a few possible exceptions at rank 12 and levels $5,7$ and $14$. Those exceptions are eliminated up to a certain bound by an extensive finite search in place of required infinite search. Our list contains a few potential unitary modular data which are not known to correspond to any unitary modular tensor categories (such as those from Kac-Moody algebra, twisted quantum doubles of finite group, as well as their Abelian anyon condensations). It remains to be shown if those potential modular data can be realized by modular tensor categories or not. We provide some evidence that all may be constructed from centers of near-group categories or gauging group symmetries of known modular tensor categories, with the exception of a total of five cases at rank 11 (with $D^2 =1964.590$) and 12 (with $D^2 =3926.660$). The classification of modular data corresponds to a classification of modular tensor categories (up to modular isotopes which are not expected to be present at low ranks). The classification of modular tensor categories leads to a classification of gapped quantum phases of matter in 2-dimensional space for bosonic lattice systems with no symmetry, as well as a classification of generalized symmetries in 1-dimensional space.