  Using M-theory in physics, Cho, Gang, and Kim (JHEP 2020, 115 (2020) )
recently outlined a program that connects two parallel subjects of three
dimensional manifolds, namely, geometric topology and quantum topology. They
suggest that classical topological invariants such as Chern-Simons invariants
of $\text{SL}(2,\mathbb{C})$-flat connections and adjoint Reidemeister torsions
of a three manifold can be packaged together to produce a $(2+1)$-topological
quantum field theory, which is essentially equivalent to a modular tensor
category. It is further conjectured that every modular tensor category can be
obtained from a three manifold and a semi-simple Lie group. In this paper, we
study this program mathematically, and provide strong support for the
feasibility of such a program. The program produces an algorithm to generate
the potential modular $T$-matrix and the quantum dimensions of a candidate
modular data. The modular $S$-matrix follows from essentially a trial-and-error
procedure. We find modular tensor categories that realize candidate modular
data constructed from Seifert fibered spaces and torus bundles over the circle
that reveal many subtleties in the program. We make a number of improvements to
the program based on our computations. Our main result is a mathematical
construction of a premodular category from each Seifert fibered space with
three singular fibers and a family of torus bundles over the circle with
Thurston SOL geometry. The premodular categories from Seifert fibered spaces
are related to Temperley-Lieb-Jones categories and the ones from torus bundles
over the circle are related to metaplectic categories. We conjecture that a
resulting premodular category is modular if and only if the three manifold is a
$\mathbb{Z}_2$-homology sphere and condensation of bosons in premodular
categories leads to either modular or super-modular categories.
