Modularity and Quantum Invariants of 3-Manifolds

Seen as complex-valued functions on rational numbers, quantum invariants of links and 3-manifolds have subtle analytic and algebraic features which seem to indicate that these topological invariants contain some information about the Riemannian structure of the underlying manifold. An example of this behavior is the quantum modularity conjecture, which is an asymptotic statement about the relation between quantum invariants whose arguments are related by a modular transformation. In this thesis, we formulate a strong version of the quantum modularity conjecture for the $\mathfrak{sl}_2$ Witten–Reshetikhin–Turaev invariant of a geometric 3-manifold. In this formulation, the quantum modularity conjecture states that for a given geometric manifold (i) the $k \to \infty$ asymptotics of the WRT invariant $\tau(k)$ contains a sum over $\mathrm{SL}(2, \mathbb{C})$ flat connections on the manifold, (ii) there is a distinguished flat connection which appears with a factor of the WRT invariant itself, its argument having undergone a modular S-transformation, $\tau(-1/k)$, and (iii) this distinguished flat connection is determined by the Thurston geometry of the manifold. This generalizes the canonical $\mathrm{SL}(2, \mathbb{C})$ flat connection on hyperbolic manifolds to other Thurston geometries. We also prove that the conjecture holds for all Seifert homology spheres with three exceptional fibers, an infinite family of rational homology spheres, and some other examples. Since the WRT invariant at a generic root of unity is morally the partition function of Chern–Simons theory at a fractional level, we also interpret this modular behavior in terms of the asymptotics of the contour path integral of analytically continued Chern–Simons theory. Finally, we discuss an ongoing work about the unification of radial limits of the $q$-series valued $\hat{Z}$-invariants into a single element of an appropriately completed ring of polynomials. Existence of such a unification for integer homology spheres, and support from our numerical experiments indicate that such a unification might exist more generally.