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I am (slowly) working on bringing the entry differential cohomology in better shape.
For the moment I am adding a section on the Hopkins-Singer formulation.
I see Ulrich's influence :) Don't forget to learn something about eta invariants and determinant line bundles (in the business of zeta function renormalization and quantum anomalies), he has a number of papers on them (I do not know what was his motivation).
That’s a good suggestion, Zoran, but I am afraid there is no time for that. But I’ll see.
I added a (very brief so far) Properties-section on the Hopkins-Singer definition.
Then I wrote on my personal web in the section General differential cohomology the outline of an argument that is supposed to show that
for $\mathcal{A}$ a topological space
and $A = LConst Sing \mathcal{A}$ its incarnation as a constant oo-stack in $\mathbf{H} = Sh_{\infty}(CartSp)$,
the Hopkins-Singer differential cohomology with coefficients in $\mathcal{A}$ coincides with the differential cohomology with coefficients in $A$ in the (oo,1)-topos $\mathbf{H}$, as defined there.
Needs more attention, but I have to run now to get some breakfast…
I am finally getting to brushing-up the entry on differential cohomology (which has been – of all entries! – woefully neglected).
For the moment I have polished and considerably expanded the Idea section and brushed-up and commented the list of references.
In particular I am splitting off an entry differential cohomology hexagon.
I have added references, but upon saving and now upon viewing the entry, it throws the following error message:
XML Parsing Error: mismatched tag. Expected: </div>.
Location: https://ncatlab.org/nlab/show/differential+cohomology
Line Number 545, Column 3:
</ul>
--^
Apparently this is caused by some code that was there before, which got re-saved now with a stricter parser. But I haven’t found the problematic line yet…
I have fixed it now. The problem came from the table of contents rendering; the problem was the use of a customised theorem name in the theorems attributed to Hopkins-Singer and Bunke-Schick, which caused the renderer to think that these theorems were sections. Unnumbered theorem environments were also being used. I have now changed the environments to numbered ones, and added the attributions to Hopkins-Singer and Bunke-Schick in a slightly different way which does not break things.
Thanks for the pointer! I hadn’t know about this.
Let’s add the author-entries you are requesting. Will do now…
there is also the ambitious “hypothesis H” of Fiorenza–Sati–Schreiber [Sat18; FSS19; FSS20b] proposing that the $C$-field in $M$-theory is quantized using a differential refinement of $Im(J)$-twisted stable cohomotopy instead.
Doesn’t need that ’stable’.
True.
Speaking of which, I am taking the liberty of adding (here) pointer to Section 4.3 in
where differential cohomology via $\infty$-sheaves is set up in the non-abelian generality that accomodates differential refinements of un-stable Cohomotopy theory.
The arxiv page says
Comments very welcome!
Yes, I was just sending them (hoping that makes sense) a comment (here) on the AT-dischord server
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