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The first Chern number $\cal C$ is known to be related to various physical objects. 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 existance of monopoles. 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 Brillion zone).

Then my questions:

  1. What was the "physical" picture in Chern's mind when he originally "dreamed up" the theory? (Maybe knots, but how?)
  2. If I want to learn how Chern classified $U(1)$ bundles using integers (first Chern number), which books or papers should I refer to?

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?

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