  A natural way to obtain a system of partial differential equations on a
manifold is to vary a suitably defined sesquilinear form. The sesquilinear
forms we study are Hermitian forms acting on sections of the trivial
$\mathbb{C}^n$-bundle over a smooth $m$-dimensional manifold without boundary.
More specifically, we are concerned with first order sesquilinear forms,
namely, those generating first order systems. Our goal is to classify such
forms up to $GL(n,\mathbb{C})$ gauge equivalence. We achieve this
classification in the special case of $m=4$ and $n=2$ by means of geometric and
topological invariants (e.g. Lorentzian metric, spin/spin$^c$ structure,
electromagnetic covector potential) naturally contained within the sesquilinear
form - a purely analytic object. Essential to our approach is the interplay of
techniques from analysis, geometry, and topology.
