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An excerpt from the book Loop Spaces, Characteristic Classes and Geometric Quantization by Jean-Luc Brylinski is mentioned below:

Characteristic classes are certain cohomology classes associated either to a vector bundle or a principal bundle over a manifold. For instance, there are the Chern classes $c_i(E)$ of a complex vector bundle $E$ over a manifold $M$. These Chern classes can be described concretely in two ways. In the de Rham theory, a connection on the bundle is used to obtain an explicit differential form representing the cohomology class. In singular cohomology, the Chern class is the obstruction to finding a certain number of linearly independent sections of the vector bundle; this is Pontryagin's original method. The relation between the two approaches is rather indirect and in some ways still mysterious. A better understanding requires a geometric theory of cohomology groups $H^p(M, \mathbb{Z})$ for all $p$.

Question :

  1. How does geometric theory (Geometric quantization) of cohomology groups $H^p(M;\mathbb{Z})$ helps in better understanding of the Chern-Weil theory approach and Singular cohomology approach (obstruction version) of characteristic classes?
  2. What more clarity are we expecting between these two versions of Chern classes?
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  • $\begingroup$ Did you try to read about Cheeger-Simons characters? In their original paper, Cheeger and Simons provide a refinement of both descriptions of Chern classes at once. $\endgroup$
    – Sebastian Goette
    Commented Apr 8, 2020 at 15:18
  • $\begingroup$ @SebastianGoette Hi.. Just to confirm, are you referring to numr.wdfiles.com/local--files/differential-cohomology/…? I have not read that, I want to read that... Can you please say in 1/2 lines in your view what refinement they are mentioning... $\endgroup$
    – Praphulla Koushik
    Commented Apr 8, 2020 at 16:23
  • $\begingroup$ Yes, that's the paper. "Differential characters" are a simultaneous refinement of (say) integral cohomology and de Rham forms (cf. Deligne cohomology in arithmetic geometry). Characteristic classes of vector bundles with connections can be refined to differential characters. Easiest example: the first Chern class of a complex line bundle with connection on $S^1$ as a differential character is equivalent to $\frac\log{2\pi i}\in\mathbb{C/Z}$ of the holonomy of the connection. Note that both the integral and the de Rham version vanish for degree reasons, so the refinement is nontrivial. $\endgroup$
    – Sebastian Goette
    Commented Apr 10, 2020 at 11:03
  • $\begingroup$ @SebastianGoette Ok. I will read and try to realte to the above mentioned question :) Thank you :) $\endgroup$
    – Praphulla Koushik
    Commented Apr 10, 2020 at 12:58
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    $\begingroup$ @SebastianGoette That is my usage. The book does not use that phrase at that point.. suppose I have a differential $2$-form $K$ on a manifold $M$, then, I can ask if this $2$-form is curvature of some connection on some line bundle over $M$.. This they called "quantization condition" in the book.. Is it not suitable word here? $\endgroup$
    – Praphulla Koushik
    Commented Apr 10, 2020 at 13:25

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