I want to switch gears and talk about some mathematical physics. Actually, I’m going to cross-post some exposition I wrote for a gauge theory seminar that we held at Stony Brook.

Maxwell’s equations in relativistically covariant form are

Since we can define a 2-form . We can also define a 1-form . Then we can re-express Maxwell’s equations using exterior differentiation and the Hodge star.

The continuity equation then follows from the inhomogeneous Maxwell equation. We expect from the homogeneous Maxwell equation that . In fact this is only true **locally**. This means that for every event in our spacetime there is an open set with and a 1-form on with . This follows from Poincare’s lemma.

We cannot say the exists globally. For instance if , the area form of the unit sphere in spherical coordinates, then since by antisymmetry of wedge product of 1-forms. Also, taking to be the unit sphere, we know that . However, by Stokes’ Theorem, if then . So, we cannot have globally.

Physically we interpret this as a magnetic monopole with magnetic charge and worldline, the time axis, . Mathematically, what is happening is that the complement of the time axis has nontrivial topology. Specifically its second de Rham cohomology is nontrivial. Intuitively, there is a kind of 2-dimensional, spherical “hole” in the complement of the time axis.

In addition to being nonglobal, the potential is defined only up to addition of a closed 1-form since . We would like to find a global mathematical object corresponding to the potential which doesn’t depend on our “choice of gauge”. This is our motivation for understanding connections on principal bundles.

We will assume is a group of matrices. A principal -bundle is a smooth surjection of manifolds with a free transitive right action of on such that and for any there is an open set with and a diffeomorphism called a “local trivialization” such that . Local trivializations correspond to the physical notion of “choice of gauge”.

Intuitively, is a manifold composed of copies of the group parametrized by the base space . A good example is the boundary of the Mobius strip which can be thought of as a -bundle over .

A useful notion is that of a local section with an open set with such that . It can be shown that there is a canonical 1-1 correspondence between local sections and local trivializations .

Define transition functions by where . This is well defined since . Transition functions correspond to the physical notion of “change of gauge”. We can relate any two local sections by .

Let be the Lie algebra for . A connection is a -valued 1-form on such that if If and is the tangent field on given by , then . Also we require that .

We define local connections on by . Local connections are related by .

We define curvature meaning . We can define local curvature by . Local curvatures are then related by . The Bianchi identity says .

We are now in a position to define electrodynamics on a principal bundle.

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