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The first Chern number $\cal C$ is known to be related to various physical objects.

  1. 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.

  2. 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:

  1. 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?

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    $\begingroup$ I thought Chern was a pure mathematician, working on the problem of classifying bundles. The applications to physical problems weren't done by him (but I may be wrong). To understand Chern classes, you need to research characteristic classes which belongs to the fields of algebraic topology/differential geometry. For this topic I always recommend this reference. $\endgroup$ – twistor59 Jan 25 '13 at 9:43
  • $\begingroup$ Thank you for the comment. 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? $\endgroup$ – Machine Jan 25 '13 at 12:31
  • $\begingroup$ The original problem that S S Chern solved was to give concrete expressions of the characteristic classes of complex vector bundles. The classes are topological, measuring how non-trivial the bundle is. Take the real line bundle over a circle as an example, the trivial bundle is a cylinder. The first non-trivial bundle is a Möbius strip. The Euler class measures how many times it twists away from the trivial cylinder. This number must be integer valued. Chern class is similar but for complex bundles. $\endgroup$ – The Last Knight of Silk Road Mar 12 '18 at 3:19
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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.

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"In quantum Hall effect(QAHE), Klitzing found the Hall conductivity to be integer multiples of a fundamental constant. This effect is independent of size and impurities of the system with which we deals with. Based on it, a famous scientist R. Laughlin proposed a theory describing the integer states in terms of a topological invariant. This topological invariant is known as chern number.

For details of the cern number,there is a wikipedia link."

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