This Thesis will focus on three different forays into particle physics using pure mathematics.

Our first foray studies anomaly free gauge algebras. Using geometric methods, we reproduce a solution given to the anomaly cancellation conditions associated with a pure u(1)-gauge theory in Costa et al. [Phys. Rev. Lett.123 (2019) 151601]. Using similar techniques, the general solution to the anomaly cancellation conditions associated with a u(1)-extension of the Standard Model gauge algebra when the chiral fermion content is that of the Standard Model plus three singlets, is found for the first time. For the same Standard Model set up, a computational approach is used to catalogue all semisimple extensions.

The second foray studies quantum mechanics in magnetic backgrounds. For such problems, it is known that a global lagrangian need not exist, and even if it does, it may shift by a total derivative under the action of the symmetry group. These two facts pose an obstruction to the standard techniques of harmonic analysis. We show that these obstructions can be overcome by passing to a redundant description with the particle moving on a U(1)-principal bundle of the original configuration space, and the symmetry replaced with an associated U(1)-central extension. We demonstrate the power of this technique using a series of examples.

For the final foray we look at the inverse Higgs phenomenon which is important for the study of Goldstone bosons. We take holonomic constraints as our starting point giving them a categoric construction. The dual of this construction leads to a new type of constraint we call a coholonomic constraint. Coholonomic constraints like holonomic ones, are equivalent to unconstrained systems. We show that every instance of the inverse Higgs phenomenon in the literature can be treated as a coholonomic constraint, or a slight generalisation thereof we call a comeronomic constraint. In this framework, the essential Goldstone bosons of the inverse Higgs phenomenon correspond to the degrees of freedom of the unconstrained system.