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have expanded the first section at determinant line bundle. Spelled out some more details and then stated the fact that the class of the determinant line bundle of a complex vector bundle is its first Chern-class, and that this statement directly refines to differential cohomology.
Hi Urs! I like that you joined the work on this important entry. I started doing this a while ago, but stayed rather cryptic. Now you are bringing some real explanations to it :)
I wanted to mention the example $det$ tomorrow for a talk on differential characteristic classes that I give in Rome, as a very simple and yet insightful example for a characteristic class and was surprised that the evident statement $[det E] = c_1(E)$ for a complex vector bundle $E$ is so hard to find explicitly in the basic literature. One finds plenty of generalized versions of this statement, but for expositional purposes I wanted just this bare one. Thanks to Domenico for tracking down references!
I have cross-linked determinant line bundle and theta function a bit more (and vacuum energy) by pointing to (Freed 87, pages 30-31)
(no time for more than just these pointers right now)
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