Topological Quantum Field Theories: gravity and finite group gauge theories

This thesis develops two applications of three-dimensional topological quantum field theories. The first concerns a toy model of topological quantum gravity defined through a sum over 3-manifold topologies with fixed boundary. By expressing the partition function of an abelian TQFT in terms of a finitely generated abelian group $A$, a non-degenerate bilinear form on $\Tor A$ and a tuple of elements of $A$, we reformulate the topological sum as a sum over algebraic data and derive sufficient conditions on the weights in the sum for convergence. Under additional assumptions, we characterize an ensemble of two-dimensional TQFTs whose averaged boundary partition functions reproduce the gravitational amplitudes, in agreement with the ensemble version of the holographic principle. For suitable weights, the analysis factorizes into independent contributions associated with each prime. The second part investigates a conjectural relation between twisted $\SL(2,\F_q)$ Dijkgraaf–Witten theory at large $q$ and $\SU(2)$ Chern–Simons theory at large level. We construct the relevant twist using second Chern classes and test the proposed relation on lens spaces, torus bundles, Seifert fibered spaces, higher-genus plumbings and other geometric 3-manifolds. The examples support a correspondence between the leading asymptotic data for non-hyperbolic geometries, while a hyperbolic example demonstrates a limitation of the strongest formulation. For spherical manifolds, we prove a weaker statement which supports the conjecture. These results illuminate interactions among topology, arithmetic and quantum theories and motivate further studies on topological quantum field theories in three dimensions.