In the first half of this talk, I will introduce recent results concerning small critical Sobolev data global well-posedness (and scattering) of the IVP for two model abelian gauge theories: massless Maxwell-Klein-Gordon (due to Krieger-Sterbenz-Tataru) and massless Maxwell-Dirac (joint work with C. Gavrus). At the heart of these results lies a parametrix construction procedure for an appropriate magnetic wave equation, which utilizes some intricate Fourier-analytic ideas. We will focus on this aspect in the second half.