Modular q–difference equations and quantum invariants of hyperbolic three–manifolds

Abstract

In this thesis, we take up the study of quantum invariants of hyperbolic closed three manifolds. In particular, we consider the behaviour of the Witten-Reshetikhin-Turaev invariants associated to the Lie algebra of the second special linear group. These invariants associate to a three-manifold a function from roots of unity to the complex numbers. Based on Witten's ideas from physics, there are various conjectures about the asymptotics of these invariants as the order of the root of unity tends to infinity. Recently, Zagier introduced the concept of quantum modular forms. In his original work, he does not gives a precise definition but a list of examples. These examples culminate in the most interesting, which came from the quantum invariants of a hyperbolic knot. In the preceding years, the idea of quantum modularity was refined and a beautiful picture for Zagier's example was built with the work of Garoufalidis, Kashaev, Gu, Mariño, Zagier with a final step taken in this thesis. We then go on to show that this holds for a closed hyperbolic three manifold. Generalising this example then unifies two seemingly different conjectures on the asymptotics of the Witten-Reshetikhin-Turaev invariant. These results are set to the back drop of the theory of q-difference equations, which give a powerful tool when constructing quantum modular forms with certain properties. Finally, we give a conjectural computation of the full resurgent structure associated to perturbative series associated to a hyperbolic closed three manifold. In particular, computing generating functions of Stokes constants. The first such example with such a precise proposal.