arXiv subjects
Publications and source records attributed to Burak Oğuz.
These notes are based on a series of lectures by Kadri İlker Berktav from May 2024 to November 2024, providing a detailed exposition of geometric quantization formalism and its essential components. They are organized into three parts: background in symplectic geometry, basics of geometric quantization formalism, and an application related to Edward Witten's work in knot theory and topology.
Performing topological manipulations is a fruitful way to understand global aspects of Quantum Field Theory (QFT). Such modifications are typically controlled by the notion of Topological QFT (TQFT) coupling across different codimensions. Motivated by the recent developments involving non-compact TQFTs as the Symmetry Topological Field Theory (SymTFT) for continuous symmetries, we realize topological manipulations on global $\mathbb{R}$ symmetries via TQFT coupling in the simple context of non-compact abelian gauge theory. Namely, by inserting the background fields for $\mathbb{R}$ symmetries into non-compact TQFTs on spacetime, we study the topological gaugings. Furthermore, we explore topological defects in non-compact theories by employing the said manipulations on the half-space, which are analogous to duality defects in compact gauge theories. We examine the action of these defects on the local and extended operators, and discuss their algebra. As opposed to the duality defects, our defects act invertibly on the spectrum. To further understand their role, we discuss the way they mix with other topological defects and the resulting global symmetries. We also comment on possible applications of these ideas to the bosonic String Theory. After studying defects, we provide a detailed inspection of manipulations on $\mathbb{R}^{(-1)}$ and $\mathbb{R}^{(d-1)}$ symmetries in abelian gauge theory. Notably, we develop a novel deformation tailored for flat gauging subgroups of $\mathbb{R}^{(d-1)}$ symmetries, and provide an extension of it for $\mathbb{R}^{(p)}$ symmetries. We show that this new simple deformation captures all the topological boundary conditions of the corresponding non-compact SymTFTs as the deformation parameter is varied, which the ordinary non-compact BF coupling cannot do.
We study symmetry reductions in the context of Euclidean Chern-Simons gauge theories to obtain lower dimensional field theories. Symmetry reduction in certain gauge theories is a common tool for obtaining explicit soliton solutions. Although pure Chern-Simons theories do not admit solitonic solutions, symmetry reduction still leads to interesting results. We establish relations at the semiclassical regime between pure Chern-Simons theories on $S^3$ and the reduced Quantum Field Theories, based on actions obtained by the symmetry reduction of the Chern-Simons action, spherical symmetry being the prominent one. We also discuss symmetry reductions of Chern-Simons theories on the disk, yielding $BF$-theory in two dimensions, which signals a curious relationship between symmetry reductions and the boundary conformal field theories. Finally, we study the Chern-Simons-Higgs instantons and show that under certain circumstances, the reduced action can formally be viewed as the action of a supersymmetric quantum mechanical model. We discuss the extent to which the reduced actions have a fermionic nature at the level of the partition function.
We study symmetry reductions in the context of Euclidean Chern-Simons gauge theories to obtain lower dimensional field theories. Symmetry reduction in certain gauge theories is a common tool for obtaining explicit soliton solutions. Although pure Chern-Simons theories do not admit solitonic solutions, symmetry reduction still leads to interesting results. We establish relations at the semiclassical regime between pure Chern-Simons theories on $S^3$ and the reduced Quantum Field Theories, based on actions obtained by the symmetry reduction of the Chern-Simons action, spherical symmetry being the prominent one. We also discuss symmetry reductions of Chern-Simons theories on the disk, yielding $BF$-theory in two dimensions, which signals a curious relationship between symmetry reductions and the boundary conformal field theories. Finally, we study the Chern-Simons-Higgs instantons and show that under certain circumstances, the reduced action can formally be viewed as the action of a supersymmetric quantum mechanical model. We discuss the extent to which the reduced actions have a fermionic nature at the level of the partition function.