The group that Weyl proposed at the beginning was noncompact and cannot preserve length. This was criticized by Einstein. But after a few years, Weyl learned from the works of London et al. in quantum mechanics that the group should be U(1). Once the group is chosen right, length is preserved under parallel transportation and [the] Maxwell equation becomes a gauge theory.
While Weyl accomplished the remarkable interpretation of Maxwell’s equations in terms of gauge theory around 1928, the theory of connections was developed by several geometers. In 1917, Levi-Civita studied parallel transport of vectors in Riemannian geometry. In 1918, Weyl in his book introduced affine connections. Cartan in 1926 studied the holonomy group for general connections.
Levi Civita and E. Cartan were interested in another approach to extend Einstein’s theory of general relativity by looking into connections with nontrivial torsion. (Einstein was using Levi-Civita connections which have no torsion.) The connection still preserves a metric. This is in fact a form of gauge theory on the tangent bundle. But Weyl’s point of view was different and he did not restrict himself to tangent bundles.
In 1944, Chern studied Hermitian connections on complex bundles and, using the curvature of the Hermitian connections, introduced the Chern classes of the bundles. They give rise to the de Rham classes of the space which turns out to be integral classes.
Upon seeing the definitions, Weil interpreted Chern’s theory in terms of invariant theory. This is called the Chern-Weil theory. It is remarkable that Weil said that at that time Chern classes may be used to quantize physical theory.
[…]
Modern development of high energy physics and theory of condensed matter shows that the prediction of Weil is accurate. In fact, not only Chern classes play an important role in modern quantum field theory, the Chern-Simons invariant, which is derived from curvature representation of Chern classes, also play an important role in condensed matter physics and string theory, which in turn influence the study of knot theory in geometry, as was shown by Witten that it can be used to explain the Jones polynomial of the knots. (Emphasis added.)
~Yau
Chern–Simons theory is an exercise in the simplicity, beauty, and weirdness of topology. It is an archetypical example of a topological field theory, a quantum field theory where the physical observables are topological invariants of the spacetime in which the theory lives. In particular, Witten showed in the late 1980s that nonlocal observables in Chern–Simons (CS) theory called Wilson loops, represented by knots in a 3-dimensional spacetime, compute certain invariants of those knots that generalize the celebrated Jones polynomial. As part of this work, which earned him the Fields medal, Witten described how to use path integrals to give a physical interpretation to such invariants. He also quantized and solved the theory, and gave an account of its Hilbert space structure. His work highlighted deep connections between diverse areas of mathematics and physics, and set off a flurry of activity that greatly enriched and brought together both fields. (Emphasis added.)
