The first Chern number $\cal C$ is known to be related to various physical objects.
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Gauge fields are known as connections of some principle bundles. In particular, principle $U(1)$ bundle is said to be classified by first Chern number. In terms of electromagnetic field, ${\cal C} \neq 0$ is equivalent to the existence of monopoles.
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In the case of integer quantum Hall states, Chern number is simply the Hall conductance up to a constant.
In both physical problems, Chern number is related to vorticity — a quantized value (first case, Dirac's string argument and second, vortices in magnetic Brillouin zone).
Then my question:
- What was the "physical" picture in Chern's mind when he originally "dreamed up" the theory? (Maybe knots, but how?)
Notes:
My point is that mathematical theorems are not God-given but arose from concrete problems. I was asking what was the original problem that Chern solved, from which he codified the general theorems? And Chern number seems related to vorticity and then what are the corresponding vortices in his problem?
Best Answer
The original reference is here (1945!). Note that before Chern classes came the Stiefel-Whitney classes, which give $\mathbb{Z}_2$ invariants of real manifolds. Chern wanted invariants of complex manifolds, so he defined his famous classes.
All-in-all, one can think of characteristic classes and their culmination, index theory, as a grand series of generalizations of the Gauss-Bonnet theorem, which gives a way of integrating a locally defined quantity (the Gaussian curvature) into a global (and quantized) topological invariant (the Euler characteristic).
Maybe you can say it's all because Gauss just wanted a better way of eating pizza.