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I've been working through some notes on differential cohomology for the past few months. I feel like I have a pretty decent grasp on the concepts and its construction, at least for differential extensions of ordinary cohomology, like Deligne cohomology.

My friend recently asked me about some of the uses of differential cohomology and I was unable to provide a good answer. What are some results of differential cohomology? Or some theorems that have been proven using these new tools from differential cohomology? I'd also be happy with some computations computed using differential cohomology. This is a general question not specific to Deligne cohomology, so answers pertaining to any differential cohomology theory (like differential K-theory) are very much encouraged.

Thanks!

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The paper

Chern, Shiing Shen; Simons, James Characteristic forms and geometric invariants. Ann. of Math. (2) 99 (1974), 48–69

seems to be seminal in the subject, and gives a flavour of how differential cohomology came about. It suggests that Chern and Simons discovered the theory "by accident", when looking for a combinatorial formula for the first Pontrjagin number of $4$-manifolds.

They soon realised that their discovery allowed certain topological invariants to be promoted to geometric invariants.

A very concrete application given in Section 5 is the use of differential Pontrjagin classes to show that $SO(3)\approx\mathbb{R}P^3$ (with the usual metric of constant curvature) does not admit a conformal immersion in $\mathbb{R}^4$.

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    $\begingroup$ As an aside, decabillionaires like Simons who have written an article in Annals of Mathematis with a mathematician of the calibre of Chern seem to form a rather select club.. $\endgroup$
    – Georges Elencwajg
    Commented Feb 20, 2013 at 11:51
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    $\begingroup$ Thank you very much for this! I was completely unaware of this application. After reading through the paper, I also understand your comment 'how differential cohomology came about'. I may just be asking too much at this point, but are there any other applications or theorems proven using differential cohomology? I mean this in the sense that, the field has continued to be worked on and developed for (over) 40 years. I think this result is very interesting but are non-conformal immersion theorems all that are able to be proven (I also don't mean to undermine the difficulty of such results)? $\endgroup$
    – Mike Catanzaro
    Commented Mar 1, 2013 at 15:56
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    $\begingroup$ @Mike: You're welcome. I must admit I don't really know much about differential cohomology, I just stumbled across this reference. Perhaps a tactfully worded email to an expert (such as Ulrich Bunke) might be in order. $\endgroup$
    – Mark Grant
    Commented Mar 1, 2013 at 16:25

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