# References for differential cohomology and differential characters

I am interested in learning differential cohomology and differential characters, and am currently studying these lecture notes on the subject. I sometimes feel it would be great if I could keep some more good references besides the lecture notes, as it could greatly help me speed up when I get stuck (which is happening often with me). Could someone please advise some alternate references which might supplement my study ?

I am familiar with elementary algebraic topology (singular homology & cohomology theories, basic homotopy theory) and differential geometry (connections, curvature, de-Rham cohomology, Chern-Weil theory).

Thanks !

Note : I earlier asked this question on m.se and offered a bounty but I did not get any answers there, I am taking the liberty of asking here.

• You might want to look at the two papers by simon and sullivan (arxiv.org/abs/0810.4935 & arxiv.org/abs/math/0701077) You can also look at the two papers (arxiv.org/abs/1507.06404 by Bunke, and arxiv.org/abs/0707.0046 by Bunke and Schick). Differential cohomology theories are designed to get invariants of flat vector bundles(there are other uses but lets stick to the easy ones) hence the most natural one to work with is differential $K$ theory so I think it is better to think about differential $K$ theory in parallel with differential cohomology
– Omar
Oct 22, 2016 at 17:28

Bunke's notes are indeed a great source for this material! However, in order to get to the main definitions and properties, he does breeze through a lot of the fundamental prerequisites. I will briefly add to the great list Omar has provided above.

For a classical, non sheaf theoretic approach: Jeff Cheeger, James Simons, Differential characters and geometric invariants. http://numr.wdfiles.com/local--files/differential-cohomology/cheeger-simons.pdf

For a great text which models differential cohomology via Deligne cohomology (and discusses the relationship with $U(1)$-gerbes in low degrees): Jean-Luc Brylinski, loop spaces characteristic classes and geometric quantization.

If you have a strong background in category theory (in particular categories of sheaves), I would strongly recommend the higher stack model of differential cohomology (similar to Brylinski's gerbes, but generalized to higher dimensions) which is discussed in: Urs Schreiber, Differential cohomology in a cohesive infinity-topos https://arxiv.org/pdf/1310.7930.pdf

For the original definition in terms of differnetial function spectra, you can see: Mike Hopkins, Isadore Singer, Quadratic Functions in Geometry, Topology,and M-Theory https://arxiv.org/abs/math/0211216

If you want to see a brief summary of Bunke's model for differential cohomology theories (the papers that Omar mentioned are far more comprehensive and thorough, but just to add a bit more to the list), there is also a short discussion in this pre-print of mine with Hisham Sati: Spectral sequences in smooth generalized cohomology https://arxiv.org/pdf/1605.03444.pdf

• Thanks so much for this very helpful answer ! I think I shall start reading Brylinski's book in parallel with the notes. One further question : In section 2.6 (p.22) of the notes, prof Bunke assumes and uses the notions of bordism classes and Thom spectra $MSO_{n}(B)$. I have not studied this stuff previously, could you please advise some reference for this ? Oct 23, 2016 at 20:13
• @user90041 Sure. Miller has some wonderful introductory notes on cobordism www-math.mit.edu/~hrm/papers/cobordism.pdf Oct 24, 2016 at 3:17
• Also, be advised that Brylinski does not use the term "differential cohomology" in that text - only "Deligne cohomology" (the latter is one particular model for the former). Oct 24, 2016 at 3:21

The following book has recently appeared:

• Araminta Amabel, Arun Debray, Peter J. Haine (eds.), Differential Cohomology: Categories, Characteristic Classes, and Connections. Based on Fall 2019 talks at MIT's Juvitop seminar by: A. Amabel, D. Chua, A. Debray, S. Devalapurkar, D. Freed, P. Haine, M. Hopkins, G. Parker, C. Reid, and A. Zhang. (arXiv:2109.12250)