MTC$[M_3, G]$: 3d Topological Order Labeled by Seifert Manifolds

We propose a correspondence between topological order in 2+1d and Seifert three-manifolds together with a choice of ADE gauge group $G$. Topological order in 2+1d is known to be characterized in terms of modular tensor categories (MTCs), and we thus propose a relation between MTCs and Seifert three-manifolds. The correspondence defines for every Seifert manifold and choice of $G$ a fusion category, which we conjecture to be modular whenever the Seifert manifold has trivial first homology group with coefficients in the center of $G$. The construction determines the spins of anyons and their S-matrix, and provides a constructive way to determine the R- and F-symbols from simple building blocks. We explore the possibility that this correspondence provides an alternative classification of MTCs, which is put to the test by realizing all MTCs (unitary or non-unitary) with rank $r\leq 5$ in terms of Seifert manifolds and a choice of Lie group $G$.