A Few Insights Into the Confinement Problem and Other Non-perturbative Questions About Gauge Theories

This thesis covers a variety of topics surrounding non-perturbative aspects of gauge theories through four published papers. The first two papers focus on domain walls in supersymmetric Yang-Mills theory (SYM) on $\mathbb{R}^3\times \mathbb{S}^1$, and are roughly split into numerical and analytical work. The numerical work investigates which domain walls are lowest energy (BPS) in $SU(N)$ SYM, verifying previous counting arguments. The deconfinement of quarks in various representations on domain walls is also studied, where it is demonstrated that the energy of two fundamental quarks on domain walls does not depend on their separation. The analytical paper studies BPS domain walls for all gauge groups, and their worldvolume topological quantum field theories (TQFTs) for $SU(N)$, $Sp(N)$, and $E_6$. For each of the TQFTs, a thorough construction of the Hilbert space is given, and it is explicitly demonstrated how the various states of the TQFT match the domain walls in SYM by examining their symmetries. Finally, deconfinement of quarks on domain walls is classified for all gauge groups. The third paper focuses on how mixed 0-form/1-form anomalies appear in the Hilbert space of gauge theories in four dimensions. Specifically, the mixed anomaly between parity and center symmetry anomaly is studied in pure Yang-Mills, and the mixed anomaly between the discrete chiral symmetry and center symmetry is studied in QCD (adj). For both theories space is taken to be a three-torus, with twisted boundary conditions acting as background fields for center symmetry. The twists reveal degeneracies in energy eigenstates at any finite volume. The last paper studies the moduli space of multifractional instantons in pure Yang-Mills on a Euclidean four-torus. Using a combination of analytical and numerical (lattice) tools, the moduli space around the constant field strength solutions of 't Hooft is explored. Specifically, the geometry is deformed away from that which admits only constant field strength solutions and the deviations therein are studied both numerically and analytically. In a different regime where constant field strength solutions are allowed it is shown that they are not the only players in the game. There, deviations from the flat solutions are studied at fixed geometry.

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