From Three Dimensional Manifolds to Modular Tensor Categories

Abstract

Using M-theory in physics, Cho et al. (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 $$\text {SL}(2,{\mathbb {C}})$$ -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 premodular 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 examples. Our main result is a mathematical construction of the modular data 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 modular data of premodular categories from Seifert fibered spaces can be realized using 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 the resulting properly premodular categories leads to either modular or super-modular tensor categories.