Yang-Mills theories are the basis for our modern understanding of particle physics, yet relatively little is known about the non-perturbative dynamics of these theories at low energies. In particular, confinement is a ubiquitous feature of Yang-Mills theory whose microscopic explanation has remained elusive since the 1960s. More recently, developments in generalized symmetries and their 't Hooft anomaly matching conditions have offered new tools for understanding these problems. During my PhD I studied non-perturbative features of center-stabilized deformations of Yang-Mills theories using a variety of analytical and computational tools. Central to my work is the use of symmetries and their anomalies, compactified spacetimes, and twisted boundary conditions. This talk will begin with a broad overview of gauge theories and confinement. I will then highlight work I have done on supersymmetric Yang-Mills theory on a compactified spacetime as a concrete example of a confinement mechanism. Following that, I will discuss in more detail recent work I have done on fractional instantons in Yang-Mills theory on a torus with twisted boundary conditions.

Fractional instantons, classical field configurations with finite action and fractional topological charge, are known to be important for confinement. Unfortunately, analytical solutions on the torus are only known when the side lengths of the torus are tuned to a specific ratio.

To better understand fractional instantons on the twisted torus with arbitrary geometry, classical lattice simulations can be employed to pin down the structure of the moduli space. I will discuss my work on these simulations, and compare my results to approximate analytical expressions, showing excellent agreement. The talk will conclude with some general remarks and outlooks on the field and future works.