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Apparently Calabi-Yau manifolds have vanishing first Chern class. Is there some simple explanation for what first Chern class of a manifold means? How to calculate it for a given metric (in physicist notation)?

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    $\begingroup$ Would Mathematics be a better home for this question? $\endgroup$
    – Qmechanic
    Commented Aug 26, 2019 at 15:41

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There are plent of books and review articles about Chern classes for physicists, but it would help to know more about what you already know before I can make recommendations. The fact that you mention an advanced topic like Calabi-Yau manifolds while not knowing about much simpler stuff like Chern classes makes me wonder what it is you are studying, and at what stage you are at. Can you give a hint? -- then I can give much better advice.

One thing is useful to know that is that Chern classes characterise line bundles, and for most types of manifolds a line bundle(a U(1) gauge field) is something extra. Calabi-Yau manifolds, however, are Kahler manifolds and they have a built-in line bundle. Do you know what a Kahler manifold is?

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  • $\begingroup$ I have studied differential geometry only for as much as was taught during a general relativity course. I'm only interested in manifolds with (pseudo-)Riemannian structure. $\endgroup$
    – Kirby
    Commented Aug 27, 2019 at 6:19
  • $\begingroup$ I don't know what a Kahler manifold is. $\endgroup$
    – Kirby
    Commented Aug 27, 2019 at 6:33
  • $\begingroup$ If that built-in line bundle is the (anti-)canonical line bundle, then all smooth manifolds have that built-in. $\endgroup$ Commented Aug 27, 2019 at 11:29
  • $\begingroup$ @Bence Racsko. I did not know that. How does it work, and does it have a Chern class? $\endgroup$
    – mike stone
    Commented Aug 27, 2019 at 12:27
  • $\begingroup$ The canonical bundle is just the exterior bundle $\bigwedge^n M$, which always exists. I don't know much about characteristic classes however, so I cannot anwer that. $\endgroup$ Commented Aug 29, 2019 at 11:39

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