String Theory on Non-Toric Singularities: D-brane Probes, Quivers, and Five-Dimensional SCFTs

This thesis investigates the interplay between geometry and supersymmetric quantum field theory, intending to develop a systematic framework for constructing and analyzing field theories associated with non-toric singular geometries. A central role is played by quiver gauge theories, which provide a bridge between geometric data and field-theoretic structures, both in the context of probe brane dynamics and in geometrically engineered theories. The first approach developed in this work examines D2-branes probing twofold and threefold geometries characterized by a background adjoint scalar field $\Phi$. This field couples to the probe via superpotential deformations of the worldvolume theory, providing a framework for the systematic derivation of three-dimensional $\mathcal{N}=2$ quiver gauge theories associated with compound Du Val (cDV) singularities—a class of non-toric geometries that arise naturally in the Higgs field construction. The second approach that we apply to these geometries is studying the five-dimensional superconformal field theories arising from M-theory reduced on these geometries. In particular, new infinite families of SCFTs are constructed from abelian orbifolds of the Reid Pagoda singularity. These constructions give rise to theories of arbitrary rank, including an infinite class of rank-one theories that can be understood as non-toric deformations of local $\mathbb{F}_2$. A distinctive feature of these models is the presence of an additional matter sector, referred to as Pagoda matter, whose vacuum expectation values obstruct the resolution of the underlying geometry. This obstruction is shown to correspond, from the field-theoretic perspective, to a mechanism termed the freezing of the gauge coupling: the Kähler modulus controlling the inverse gauge coupling is dynamically forced to vanish, preventing access to a weakly-coupled regime and rendering the theory intrinsically strongly coupled. The origin of this phenomenon is traced to the interplay between Higgs field backgrounds and dynamical geometric deformations. More broadly, the results of this thesis suggest that non-toric geometries support a richer structure of quantum field theories than previously understood, and that additional geometric data, not captured by the naive local description, may play a role in determining the physical content of the theory.