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if you have been looking at the logs you will have seen me work on this for a few days already, so I should say what I am doing:
I am working on creating an entry twisted smooth cohomology in string theory . This is supposed to eventually serve as the set of notes for my lectures at the ESI Program K-Theory and Quantum Fields in the next weeks.
This should probably sit on my personal web, and I can move it there eventually. But for the moment I am developing it as an $n$Lab entry because that saves me from prefixing every single wiki-link with
nLab:
I have been further working on twisted smooth cohomology in string theory, it should be taking shape now. The first third plus a bit more should be readable, and a skeleton of the rest should be visible now.
Just in case anyone feels like giving feedback… ;-)
One small thing (not really related to the big picture). This remark
the left morphism is stalk-wise (around small enough neighbourhoods of each point) an equivalence of groupoids;
made the idea that anafunctors are a localisation wrt to the local equivalences (from the simplicial sheaves pov) a whole lot clearer. I will use this slogan at some point.
Thanks for the feedback!
Yes, I used to give in talks a more technical description of the localization, when at some point I realized that this here is maybe the most direct way to understand the definition for the purposes of a talk (if maybe not the most useful way to think about the localization for purposes of actually working with it).
When physicists say expression “$p$-form gauge theory” do they necessarily mean connections for $p$-gerbes and when they say “Abelian $2$-form gauge theory” the connections for $U(1)$-2-gerbes ?
In the old literature this was certainly not the case. For instance there are famous articles on BV-quantization of $p$-form fields which just considered $p$-forms, nothing else. Nowadays this is changing. Slowly.
Is there any kind of pattern to the table? Is there any attempt at exhaustiveness? Are these just a sample of physically interesting cases?
Yes, that’s a great question.
First I would like to amplify that the fact that I can tabulate this way at all means that already quite a bit of pattern has been identified. I found that and keep finding that quite pleasing.
One evident thing that I currently don’t make explicit in the table is supergeometry. That gives natural explanations for some of the structure seen there.
But of course we should never be satisfied: there must be even deeper patterns at work.
Right now I cannot quite give you a formal candidate for such, but I do have various guesses. There is accumulating evidence (say in the recent arXiv1112.3989) that the table entry corresponding to the coefficient bundles
$\array{ \mathbf{B} H_n \\ \downarrow \\ \mathbf{B}E_{n(n)} }$somehow contains in it the seed for all the other entries in the table. That would be nice, since it would anchor the whole phenomenon in just exceptional models in higher Lie theory. But I don’t fully understand many details of this. I’ll try to keep my eyes open, though.
the coefficient bundles…somehow contains in it the seed for all the other entries in the table.
How does the notion of ’containing a seed’ express itself in arrow language?
My reading group has started on Hegel’s Shorter Logic. I start to see what interested Lawvere:
Every human being is an entire world of representations buried in the night of the ’I’. The ’I’ is thus the universal in which abstraction is made from everything particular, but in which at the same time everything lies shrouded. It is therefore not a merely abstract universality, but a universality that contains everything within itself. (Sec 24)
Homework: formulate this arrow-theoretically.
I wrote:
[…] somehow contains in it the seed for all the other entries in the table. [..] But I don’t fully understand many details of this
David asks:
How does the notion of ’containing a seed’ express itself in arrow language?
That’s precisely what I don’t understand yet.
To be more explicit, the hints that exceptional generalized geometry “somehow contains the seeds” for all the other higher structures is that the exceptional generalized vielbein fields contain all the higher differential form field data. But just as form field data, not as the required higher cocycles. Currently in the literature this is being pieced together “by hand”. What is missing is a systematic picture.
I find myself increasing confused about the abstract general vs concrete particular distinction. Isn’t it the case that some concrete particulars contain something general about them? Perhaps easy instances are special universal concrete particulars, such as initial objects, but then maybe your ’seed containing’ entity is another.
How should I think of the 6d (2, 0)-supersymmetric QFT story? The QFT is a concrete particular, with two particular compactifications retaining a particular duality (S-duality) from their parent? And then
geometric Langlands duality is just an aspect of a special case of this.
Everything sounds very particular, and yet out of this comes something sounding quite general - the geometric Langlands correspondence .
Everything sounds very particular, and yet out of this comes something sounding quite general - the geometric Langlands correspondence.
It’s still pretty clearly a particular, I think, with those choices of classes of groups, and of very specific moduli stacks built from them, and very specific “2-linear maps” over them.
added a brief paragraph twisted smooth cohomology – Further twists – Relative fields with brief remarks on how the definitions of “relative fields” in the recent arXiv:1212.1692 are special cases of the general notion of twisted smooth cocycles discussed there.
Currently I am at a meeting titled Twists, generalised cohomology and applications.
I’ll be speaking on Tuesday. My talk notes (or the preliminary version of them, at least) I have added as a second main section – section B – to the page with the lecture notes at the ESI last year, so they are now here:
I had announced a talk on Local prequantum field theory (schreiber) and its relation to twisted cohomology. But now discussion here and recently made me change my plans a bit. Because, suddenly everybody is talking about $\infty$-stacks on the site of manifolds. I overhear over lunch statements like “Oh, I spent last month thinking about the $\infty$-stacks on the site of smooth manifolds”. Also the stabilized version is being pushed now, and sometimes I hear the sentiment voiced that this is crucially a different story than unstable cohesion.
I thought if many people are interested in understanding this right now, then I’d adapt a bit and highlight some basics of cohesive cohomology a bit, rather than concentrating all on its application to local prequantum field theory. That will now just be an outlook at the end of my talk.
Now we know about tangent stabilizing, can we make sense of that Goodwillie extract cited at Goodwillie calculus?
Rhetorical question: If the first derivative of the identity is the identity matrix, why is the second derivative not zero? Answer: Some of the terminology of homotopy calculus works better for functors from spaces to spectra than for functors from spaces to spaces. Specifically, since “linearity” means taking pushout squares to pullback squares, the identity functor is not linear and the composition of two linear functors is not linear.
Attempted cryptic remark: Unlike the category of spectra, where pushouts are the same as pullbacks, the category of spaces may be thought of has having nonzero curvature.
Correction: After the talk Boekstedt asked about that remark. We discussed the matter at length and found more than one connection on the category of spaces, but none that was not flat. In fact curvature is the wrong thing to look for. There are in some sense exactly two tangent connections on the category of spaces (or should we say on any model category?). Both are flat and torsion-free. There is a map between them, so it is meaningful to subtract them. As is well-known in differential geometry, the difference between two connections is a 1-form with values in endomorphisms (whereas the curvature is a 2-form with values in endomorphisms). Thus there is a way of discussing the discrepancy between pushouts and pullbacks in the language of differential geometry, but it is a tensor field of a different type from what I had guessed.
If analogy is anything to go by here, then it should be like this:
the first derivative of a functor between $\infty$-toposes
$f \;\colon\; \mathbf{H}_1 \longrightarrow \mathbf{H}_2$should be a functor between tangent $\infty$-toposes
$d f \;\colon\; T\mathbf{H}_1 \longrightarrow T\mathbf{H}_2 \,.$That and only that is the statement which is true for ordinary derivatives.
While I haven’t thought much at all about Goodwillie calculus yet, clearly you are right that I should.
Since I just pointed someone to the section relative fields, that note
Examples and further details are discussed in Schreiber, section 4,
needs updating. Is the best place section 7.1?
Probably.
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