This dissertation aims to study quantum anomalies and some other aspects of the Standard Model of Particle physics. In any quantum gauge field theory, anomalies place a very restrictive condition on the matter content and the dynamics. The former is due to the cancellation of gauge anomalies while ’t Hooft anomaly matching constraints produce the latter. As the Standard Model, which is our most fundamental and most accurate description of particle physics, is constructed as a gauge field theory, it is also subject to these anomalies. Here we explore subtleties in anomalies that could arise from the Standard Model and also use them to provide a consistency check as we explore its phase diagram.

We start by reexamining local anomaly cancellation in the Standard Model. It has long been known that the requirement that all gauge anomalies and the mixed gauge-gravitational anomaly cancel lead to the quantisation of hypercharge and essentially give the unique hypercharge assignment to the fermion content of the theory. However, if we take the view that hypercharge must be quantised from the outset, then it is enough to prove that the fermions have the Nature-assigned hypercharges using the cancellation of gauge anomalies alone. This remarkable result is made more astounding by the fact that Fermat’s Last Theorem plays a crucial role in completing the proof.

We then move on to search for subtler global anomalies in the Standard Model and beyond from the modern viewpoint of cobordism theory, where a global anomaly can be computed as a homomorphism from a bordism group of manifolds equipped with appropriate spin and gauge bundle structure to a circle group. Since the gauge interaction depends on the gauge group G only through its Lie algebra, there are many possibilities for the gauge group of a gauge theory as long as the global structure is consistent with the matter content. In the Standard Model, the options for the gauge group G are U(1) × SU(2) × SU(3), U(2) × SU(3), SU(2)×U(3), or U(2)×SU(3)/Z₃. We compute the fifth spin-bordism group of manifolds equipped with these G-bundle structures Ω₅Spin(BG) and show that it is at most Z₂. Therefore, the global anomaly that can appear in the Standard Model is a mod 2 anomaly which can be identified with the well-known Witten anomaly in the gauge group SU(2). We repeat the bordism group calculation for some beyond the Standard Model gauge groups and obtain a similar result: there is a mod 2 anomaly whenever there is an SU(2) factor in the gauge group.

A curious fact from these bordism calculations is that the bordism group is trivial when U(2) appears in lieu of SU(2). Driven by this curiosity, we investigate further and find that there is an interplay between the local and the global anomaly. The condition for the gauge anomaly cancellation on the SU(2)-representations of the fermions coupled to a gauge theory is the same whether the gauge group is SU(2) or U(2). However, the condition comes from the cancellation of the global Witten anomaly in the former case while it arises from the mixed anomaly cancellation between the U(1) sector and SU(2) sector in the latter case. We investigate further to see whether we can give the same interpretation to the new SU(2) anomaly of Wang, Wen, and Witten when we place a U(2) gauge theory on a non-spin manifold. We find that even though the requirement that the mixed gauge and the mixed gauge-gravitational anomalies cancel automatically cancel the new SU(2) anomaly, it cannot be thought of as arising from the local anomalies. The reason is essentially because the transformation that induces the new SU(2) anomaly involves a non-trivial diffeomorphism on the underlying manifold. Mathematically, we can compute the bordism group and still see a factor of Z₂ associated with this new global SU(2) anomaly.

Finally, we turn our attention towards the Standard Model itself, leaving anomalies as a tool we occasionally use to provide a consistency check on the IR dynamics. We apply the philosophy that one can get information and intuition on a theory by studying a collection of theories in the parameter space to the Standard Model. In these variations of the Standard Model, we deviate the Yukawa couplings from the actual values so that they are insensitive to the generations of fermions. We then vary the relative strength between the strong nuclear force and the weak nuclear force and see what happens. The results are surprising. No phase transition seems to be present when there is only one generation of fermions. More remarkably, the leptons seem to smoothly mutate into quarks as we slowly dial the relative strength between the weak and the strong gauge group. When more than one generations of fermions are present, however, the global symmetry group on either end of the phase diagram is not a subgroup of the other, and a first-order phase transition is expected to occur.