  We present a novel M-theoretic approach of constructing and classifying
anyonic topological phases of matter, by establishing a correspondence between
(2+1)d topological field theories and non-hyperbolic 3-manifolds. In this
construction, the topological phases emerge as macroscopic world-volume
theories of M5-branes wrapped around certain types of non-hyperbolic
3-manifolds. We devise a systematic algorithm for identifying the emergent
topological phases from topological data of the internal wrapped 3-manifolds.
As a benchmark of our approach, we reproduce all the known unitary bosonic
topological orders up to rank 4. Remarkably, our construction is not restricted
to an unitary bosonic theory but it can also generate fermionic and/or
non-unitary topological phases in an equivalent fashion. Hence, we pave a new
route toward the classification of topological phases of matter.
