This thesis explores the nonperturbative physics of three-dimensional gauge field theories with varying amounts of supersymmetry through an analysis of their one-dimensional topological substructures.

In the setting of four supercharges, we consider the topological quantum mechanics of Wilson lines in pure Chern-Simons theory. We reframe the perturbative renormalization of observables in this theory in terms of a localization principle associated with an underlying N = 2 supersymmetry. This perspective allows the otherwise perturbative corrections to be interpreted as nonperturbative consequences of a non-renormalization theorem.

In the setting of eight supercharges, we develop an approach to the study of Coulomb branch operators in 3D N = 4 gauge theories and the deformation quantization of their Coulomb branches. To do so, we leverage the existence of a 1D topological subsector whose operator product expansion takes the form of an associative and noncommutative algebra on the Coulomb branch. For "good" and "ugly" theories in the Gaiotto-Witten classification, we exhibit a trace map on this algebra, which amounts to a procedure for computing exact correlation functions of a class of local operators, including certain monopole operators, on S^3. We introduce a "shift operator" formalism for constructing correlators on S^3 by gluing hemispheres HS^3, and we show how to recover our results by dimensionally reducing the line defect Schur index of 4D N = 2 gauge theories. We use our results to study 3D mirror symmetry and to characterize monopole bubbling effects in nonabelian gauge theories. In the process, we arrive at a physical proof of the Bullimore-Dimofte-Gaiotto abelianization description of the Coulomb branch.