Well, since there are lots of discussions on algebraic topology, after the realization of topological insulators in condensed matter, an important concept I’ll try ellaborate here, which is the concept of fiber bundles.
I’ll try to control myself as less algebraic as possible. Let me first mention, what is going inside the quantum Hall market. There are two kind of businessmen. The businessman of the first kind (may be his name is Laughlin) will say, “It’s just a matter of gauge invariance”. The other businessmen (Look, they are four, perhaps their names are Thouless, Kohmoto, Nightingle, and Nijs. You can call them TKNN together in short.) will say, “The integer corresponds to the first Chern class of a $U(1)$ principal fiber bundle on a torus geometry.”
Today we’ll try to comprehend what the businessmen of the second kind wanted to mean.
So I shall first start with the theory of fiber bundles. The theory is a generalization of the direct product concept. Now what is a direct product?
Let me give you a task first. You have been given two circular plates: disk1 and disk2. Can you connect those plates by threads of equal lengths and make a solid cylinder? If you look at the picture of the ligament (a tissue connecting bones), you can easily see how it is possible. The fiber like structures inside the ligament is known as fiber bundles in biology.
Now you can write this job in a mathematical way as follows.
Now you can make a hollow cylinder in the same way by threading two hollow discs, viz. ring1 and ring2:
Now we can always find a projection that maps disk1 to disk2 in the first example or ring1 to ring2 through the threads (lines, 1D in Euclidean real space) . We would like to call disk1 or ring1 as the base space B and
the projected space (disk2 or ring1) as total space E. And from the ligament analogy, the threads will be also called as fibers F.
Thus we can write: .
We may find a projection from E to B given
Note that in our example E and B are differentiable manifolds and F is
a topological space. No need to worry about definitions of those terms. Just remember that all of them are subsets of Euclidean space, which is are most familiar differentiable manifold). Now in addition to this, to form a fiber bundle we need a few more conditions. But I won’t go into that to avoid complication. But overall these topological spaces form a fiber bundle in presence
of a Lie group G operating on the fiber F from. And we often denote the fiber bundle as or just .
Now you can see that we can map coordinates from the base space to the total space or vice-versa through fibers.
In the ring example, if we construct a hollow cylinder, the fiber bundle will be trivial. However, if we keep a twist, i.e. form
a Moebius strip, then the fiber bundle will be non-trivial. The presence of twists gives a characteristic that decides the nontrivial
topological invariant, e.g. the Chern numbers (Will be discussed soon).
Ok, now don’t you see the fibers? ! Look at the clips below.
Now what about the quantum Hall effect? This will be discussed in Part-II .