Quantum topology and modularity

Mathematics is an extremely powerful tool in physics. The effectiveness of mathematics in physics has driven vivid discussions on whether mathematics is just a tool in physical research or if there is something fundamentally mathematical about Nature itself. This thesis presents advances made towards various mathematical problems following a paradigm inversion between physics and mathematics: one where physics is used as a tool for mathematical research.
The central object of discussion is the Ẑ-invariant, a quantum invariant for 3-manifolds which is physically constructed as the half-index of a 3D reduction of a 6D N=(2,0) M-theory. The Ẑ-invariant provides bridges connecting physics to disparate disciplines of otherwise disconnected disciplines of mathematics, including Low dimensional and Quantum Topology, Modularity and Vertex Operator algebras. Following an introduction and a background chapter, where all the necessary prerequisites for the remaining chapters are presented, this thesis presents advances made towards the elucidation of the structures inherent in these connections. Chapter 3 focusses on the connections between the Ẑ-invariant and log-VOAs. It is found that Ẑ-invariants for certain families of 3-manifolds may be regarded as linear combinations of characters of log-VOAs. Chapter 4 generalizes the known modular properties of Ẑ-invariants. It is shown that classes of Ẑ-invariants for certain families of 3-manifolds are higher rank quantum modular forms. In Chapter 5 defect Ẑ-invariant are introduced as well as it discusses their modular properties. Moreover, previously unknown Ẑ-invariants are proposed which are shown to relate to characters of cone-VOAs.