# Mathematical structure of gauge theories

What physical motivations lead to the mathematical structure of gauge theories? (principal bundles, connections, ...). I know it's a very vague question, I'm looking for some concrete examples to try to understand how you get to that mathematical structure

A lot of times in theoretical physics, there is a gap between the theoretical and mathematical framework, and the underlying physical assumptions.

Before getting to gauge theory, let's consider the idea of curved spacetime (which is actually quite similar but involves a less abstract space so is perhaps a bit easier to think about). What physical motivation leads one to postulate that we should describe space and time as a pseduo-Riemannian manifold? Well the physical motivation doesn't really lead you directly to that. Historically, the logic was something like:

• Maxwell's equations were discovered.
• People noticed that Maxwell's equations were Lorentz invariant (and discovered the Lorentz transformations).
• Einstein elevated the Lorentz transformations by pointing out how they arise from the principle of inertia.
• Minkowski realized that there was much more elegant formulation of Lorentz transformations by describing spacetime as a manifold with the Minkowski metric.
• Generalizing a Minkowski metric to a general metric with the same signature leads to the idea of a curved spacetime.
• Differential geometry turns out to be exactly the formalism Einstein needed to describe gravity. The equivalence principle is included very naturally in the idea from differential geometry of considering the metric in a tangent space of a point.
• Consistency requirements (recovering Newtonian gravity, having a conservation of stress-energy, limiting the number of derivatives in the equations of motion to two) led to the correct dynamical equations the metric should satisfy (Einstein's equations).

Of course this list is greatly simplified. But the point is that one doesn't jump directly from a physical idea to the full, elegant formalism. One builds up in stages. While the chain of reasoning beings with a physical idea, some of the later steps are purely mathematical "cleaning up" of the concepts into a more elegant framework. Sometimes there is even information that flows in the opposite way, where a mathematical generalization that seems clear mathematically but doesn't have a physical motivation, ends up being relevant for physics.

In the case of Yang-Mills theory -- with the caveat that I am not a historian of science and I am not claiming this is the historical development, just one logical way of piecing together the concepts I personally like -- I would describe the chain of reasoning as

• Maxwell's equations were discovered.
• People realized you could fruitfully describe the equations using a gauge field (vector and scalar potential).
• In developing quantum electrodynamics, people realized the gauge potential allowed for a much more elegant way of formulating the quantum theory than the electric and magnetic fields.
• People realized that the vector potential could be described mathematically as a connection for a $$U(1)$$ symmetry.
• Yang and Mills had the idea to consider the connection for more general groups like $$SU(N)$$ -- this was a purely mathematical idea with no obvious physical reasoning behind it.
• It was eventually realized this formalism could be applied to describe the strong and weak interactions (again this was not at all obvious when Yang-Mills theory was first written down). The key developments here were the discovery of the Higgs mechanism (relevant for the weak interactions) and of asymptotic freedom in QCD (strong interactions).
• Along the way, given the interest in $$SU(N)$$ gauge theories, people found alternative and elegant mathematical ways of describing the theory in terms of bundles, connections, and so on. Again I'm not a historian of science, but my impression is that physicists reinvented many concepts that had been studied by mathematicians decades earlier, and so part of this process involved people bridging this gap and cleaning up or enhancing the physics formulation with previously discovered mathematical ideas.

So I think your question is not quite in the right direction, since the arrow from a physical idea to a mathematical formalism often involves many intermediate stages and even goes in directions you don't expect. But, I think it's still an interesting question to think about, and I tried to highlight this interplay between math and physics (particularly the information flow from math to physics) that isn't always appreciated.