TM Theories and Quantum Topology: Enlarged Moduli Spaces, Categorification, and Modularity

This thesis studies the Ẑ-invariants of three-manifolds, q-series valued invariants which are half-indices of 3d N=2 theories TM₃ obtained by compactifying the 6d (2,0) theory on a three-manifold. These invariants sit at the intersection of supersymmetric field theory, low-dimensional topology, and number theory. We develop their structure along four directions. A unifying theme is that the natural structures associated with Ẑ are organized by enlargements of the usual moduli spaces appearing in Chern-Simons theory. First, we construct the Ẑ-TQFT, a generalized topological quantum field theory that computes Ẑ. Unlike Atiyah-type TQFTs, this framework involves infinite-dimensional Hilbert spaces and depends on Spinᶜ-structures. A central ingredient is a Q-extended quantization of SL(2C) Chern-Simons theory on the torus in which the space of flat connections (C/Z)² is replaced by C². Second, we propose a combinatorial categorification of WRT invariants. The resulting homologies have Ẑ as their graded Euler characteristic, whose weighted sum over Spinᶜ-structures recovers the WRT invariant. These homologies are computable for a large class of three-manifolds and provide a prototype for the physically defined but largely inaccessible BPS cohomologies of TM₃. Third, we study the quantum modularity of Ẑ-invariants with line defects. The half-indices with different line defects organize into a module P over the ring of polynomials in q, and we uncover a connection between P and a module of certain meromorphic functions that elucidates the algebraic structure of P. Finally, we introduce ceff, an effective central charge measuring BPS degrees of freedom in 3d N=2 theories. When applied to TM₃, this quantity yields a new topological invariant that provides a particular integer lift of the Chern-Simons invariant. We use ceff as a probe to test competing proposals for extending Ẑ beyond the class of weakly negative-definite plumbed three-manifolds, finding that the resurgence-based and regularized-surgery approaches yield different results.