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!



Results from Differential Cohomology
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 $\mathbb{R}^4$.


