Building a Toy Principle Bundle: Visualizing Connections on Manifolds

[selfAdjoint]

In this thread I am going to build a principle bundle based on the torus. The advantage of this ultra simple case is that everything can be visualized. The disadvantage is that some things are really too simple to convey what needs to be conveyed. When I hit one of those points I will add some extra comment in bold face to try to cover the problem.

There is one other thing. I am going to start a second thread on this boqard to hold your comments. Please be kind to those reading about the model by putting all comments and criticisms of my development on that thread rather than this one. It makes it hard to read when there are many posts inline. If your ideas turn out better than mine I can always edit the posts on this thread.

Before I introduce the toy model in the next post, let me say a bit about principle bundles. We have had some discussions recently about connections and metrics on manifolds. Principle bundles give a way to introduce connections on manifolds without using metrics. The thing that does it is the connection 1-form and we will see how this arises.

Principle bundles are an example of a more general structure called a fiber bundle. General fiber bundle theory is a large topic and an important part of differntial geometry. Principle bundles, as the name might imply, are central to that theory because they can be used to construct other bundles. We won't go into that aspect of them here.

I will be introducing the various "parts" of a fiber bundle as well as a principle bundle in the next post on this thread.

 

[selfAdjoint]

Introducing the Model

Visualize a torus. Not a solid one, but just the shell. "The sugar on the donut", or an inflated inner tube without a valve.

This torus is the cartesian product of two circles, just as the euclidean plane is the cartesian product of two lines. By this I mean that every point on the torus can be represented by two coordinates, both of which are points on a circle, angles in radians or degrees. One coordinate gives a location on the big circle that goes around the hole in the torus on the outside, where the tread would be on a tire. Some point is fixed as 0, in degrees or radians, and a direction is fixed as +, and then every point on that big circle is identified by and angle from 0, +-wise.

Once you have figured the first coordinate, and are at some nice generic point on the big circle, you get the second coordinate from the little circle that goes through the same point and passes through the hole in the torus. That coordinate is just another angle, figured the same way as the first. So as I said above, every point on the torus is represented by a pair of coordinates, one from each of two circles, and that's what you mean by the cartesian product of two circles.

----Check your understanding, How can you tell that the sphere isn't the product of two anythings?--------------

Now I'm going to make a copy of the big circle and put it down below the torus (visualize). This circle is going to play the part of the manifold in the bundle. It is a manifold, a one dimensional one. We will refer to it as the base space or the base manifold or just the manifold. We will desgnate it by M and its generic point by x. Because its a standin for any old manifold, we won't much use any special properties it has as a circle. The important thing is there's a one-to-one point map from the big cirlce on the torus to the base manifold.

Now each of the little circles of the torus passes through just one point of the big circle, and so there is a function that maps each little circle into just one point on the base manifold. This function is called the projection. It maps the torus onto the base manifold, and the preimages of points in the manifold are all alike (all circles in this case. The kind of 'alike' that applies in fiber bundle theory is topological equivalence,aka homeomorphism).

This is a special kind of fiber bundle called a trivial bundle, because it's just a cartesian product. This is part of what makes our model a toy, we are not going to discuss nontrival bundles here. In a more general bundle the things we are going to do would be done in the part of the bundle that lies over an open set in the manifold and is a cartesian product in that limited domain

Some notation. The torus will be called the bundle space and denoted P, the fibers (little circles) will be denoted G (we'll see why in the next post). A typical point in a fiber G will be called g. We have already pointed out that any point in P can be representd by (x,g) where x is a point of the base manifold M and g is a point of the fiber over x, G_x.

 

[selfAdjoint]

Each Fiber is a Group

The circle is actually a group, the unitary group in one (complex) dimension. To see this we look upon the plane as the field of complex numbers, and the unit circle in that plane, centered at the origin and having radius 1, is given by the equation z = e^it, Here t, a real number, specifies the angle.

Now to rotate a complex number through an angle s, you multiply it by e^is, so we see that each point on the circle corresponds to a transformation of the plane - a rotation through an angle equal to the coordinate of the point.

Furthermore these transformations are all unitary. In the language of complex matrices, the adjoint of a given matrix is the matrix you get by transposing it - changeing its rows to columns and vice versa, and then converting all the complex entries into their complex conjugates: x + iy -> x - iy. Now e^is is just a number, no rows and columns to switch, so its adjoint is defined to be its conjugate. And the conjugate of e^is is e^-is.

The definition of unitary is that the product of every transformation with its adjoint gives unity, or, saying the same thing, the inverse of every unitary matrix is its adjoint. In this case if we multiply e^ise^-is = e^is-is = e⁰ = 1. So these transformations, points of the circle, are unitary. They form the group U(1) which is the gauge group of electromagnetism. And this group is isomorphic to the circle.

The circles of which this property is to hold in our model are the fibers. The "little circles" of the torus, which are projected down into single points on the base manifold M. So over every point in M, in this fiber bundle, lies a Lie group, U(1). This is the definition of a principle fiber bundle, a fiber bundle in thich the fibers are a group.