In physics we have the concept of a vector field. Intuitively, a vector field is given by specifying a vector (in the sense of a quantity with magnitude and direction) at every point in a certain “space”. For instance, the wind velocity on the surface of our planet is a vector field. If we neglect the upward or downward dimension, and look only at the northward, southward, eastward, and westward directions, we have what we usually see on weather maps on the news. In one city the wind might be blowing strongly to the north, in another city it might be blowing weakly to the east, and in a third city it might be blowing moderately to the southwest.
If, instead of specifying a vector space (see Vector Spaces, Modules, and Linear Algebra) at every point, instead of just a single vector, we obtain instead the concept of a vector bundle. Given a vector bundle, we can obtain a vector field by choosing just one vector in the vector space. More technically, we say that a vector field is a section of the vector bundle.
A vector space can be thought of as just a certain kind of space; in our example of wind velocities on the surface of the Earth, the vector space that we attach to every point is the plane endowed with an intuitive vector space structure. Given a point on the plane, we draw an “arrow” with its “tail” at the chosen origin of the plane and its “head” at the given point. We can then add and scale these arrows to obtain other arrows, hence, these arrows form a vector space. This “graphical” method of studying vectors (again in the sense of a quantity with magnitude and direction) is in fact one of the most common ways of introducing the concept of vectors in physics.
If, instead of a vector space such as the plane we generalize to other kinds of spaces such as the circle , we obtain the notion of a fiber bundle. A vector space is therefore just a special case of a fiber bundle. In Category Theory, we described the torus as a fiber bundle, obtained by “gluing” a circle to every point of another circle. The shape that is glued is called the “fiber“, and the shape to which the fibers are glued is called the “base“.
Simply gluing spaces to the points of another space does not automatically mean that the space obtained is a fiber bundle, however. There is another requirement. Consider, for example, a cylinder. This can be described as a fiber bundle, with the fibers given by lines, and the base given by a circle (this can also be done the other way around, but we use this description for the moment because we will use it to describe an important condition for a space to be a fiber bundle). However, another fiber bundle can be obtained from lines (as the fibers) and a circle (as the base). This other fiber bundle can be obtained by “twisting” the lines as we “glue” them to the points of a circle, resulting in the very famous shape known as the Mobius strip.
The cylinder, which exhibits no “twisting”, is the simplest kind of fiber bundle, called a trivial bundle. Still, even if the Mobius strip has some kind of “twisting”, if we look at them “locally”, i.e. only on small enough areas, there is no difference between the cylinder and the Mobius strip. It is only when we look at them “globally” that we can distinguish the two. This is the important requirement for a space to be a fiber bundle. Locally, they must “look like” the trivial bundle. This condition is related to the notion of continuity (see Basics of Topology and Continuous Functions).
The concept of fiber bundles can be found everywhere in physics, and forms the language for many of its branches. We have already stated an example, with vector fields on a space. Aside from wind velocities (and the velocities of other fluids), the concept of vector fields are also used to express quantities such as electric and magnetic fields.
Fiber bundles can also be used to express ideas that are not so easily visualized. For example, in My Favorite Equation in Physics we mentioned the concept of a phase space, whose coordinates represent the position and momentum of a system, which is used in the Hamiltonian formulation of classical mechanics. The phase space of a system is an example of a kind of fiber bundle called a cotangent bundle. Meanwhile, in Einstein’s general theory of relativity, the concept of a tangent bundle is used to study the curvature of spacetime (which in the theory is what we know as gravity, and is related to mass, or more generally, energy and momentum).
More generally, the tangent bundle can be used to study the curvature of objects aside from spacetime, including more ordinary objects like a sphere, or hills and valleys on a landscape. This leads to a further generalization of the notion of “curvature” involving other kinds of fiber bundles aside from tangent bundles. This more general idea of curvature is important in the study of gauge theories, which is an important part of the standard model of particle physics. A good place to start for those who want to understand curvature in the context of tangent bundles and fiber bundles is by looking up the idea of parallel transport.
Meanwhile, in mathematics, fiber bundles are also very interesting in their own right. For example, vector bundles on a space can be used to study the topology of a space. One famous result involving this idea is the “hairy ball theorem“, which is related to the observation that on our planet every typhoon must have an “eye”. However, on something that is shaped like a torus instead of a sphere (like, say, a space station with an artificial atmosphere), one can have a typhoon with no eye, simply by running the wind along the walls of the torus. Replacing wind velocities by magnetic fields, this becomes the reason why fusion reactors that use magnetic fields to contain the very hot plasma are shaped like a torus instead of like a sphere. We recall, of course, that the sphere and the torus are topologically inequivalent, and this is reflected in the very different characteristics of vector fields on them.
The use of vector bundles in topology has led to such subjects of mathematics such as the study of characteristic classes and K-theory. The concept of mathematical objects “living on” spaces should be reminiscent of the ideas discussed in Presheaves and Sheaves; in fact, in algebraic geometry the two ideas are very much related. Since algebraic geometry serves as a “bridge” between ideas from geometry and ideas from abstract algebra, this then leads to the subject called algebraic K-theory, where ideas from topology get carried over to abstract algebra and linear algebra (even number theory).
Geometrical Methods of Mathematical Physics by Bernard F. Schutz
Geometry, Topology and Physics by Mikio Nakahara