Splitting the Lorentz algebra into su(2) ⊕ su(2), labelling representations by (j+, j-), and the Weyl, Dirac and Majorana spinors that come out of it.

The Lorentz group O(3,1): its defining equation, its four disconnected components, its generators and its basic representations.

Lie groups and Lie algebras from the ground up: generators, structure constants, representations, Casimir operators and the exponential map.

Deriving the Faddeev-Popov identity starting from ordinary integrals, then using it to get the gauge-fixed photon propagator and to see which gauge-fixing terms are allowed.

Why the path integral for a gauge field doesn't work as written, and the geometric picture behind gauge fixing in the abelian case.

Perturbations of the massive vector and the massless 2-form: which modes propagate, and how they could show up in observations.

Putting a massive vector and a massless 2-form on an expanding FLRW background and checking whether they can drive accelerated expansion.

Gauge symmetry, equations of motion, constraints and degree-of-freedom counting for a massive vector and a massless 2-form.