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October 30, 2008

Twisted Differential Nonabelian Cohomology

Posted by Urs Schreiber

This is something we are currently working on, various aspects of which have been the subject of recent discussion here:

Hisham Sati, U. S., Zoran Škoda, Danny Stevenson
Twisted differential nonabelian cohomology
Twisted (n1)(n-1)-brane nn-bundles and their Chern-Simons (n+1)(n+1)-bundles with characteristic (n+2)(n+2)-classes
(pdf, 60 pages theory, 40 pages application currently (but still incomplete))

Abstract. We introduce nonabelian differential cohomology classifying \infty-bundles with smooth connection and their higher gerbes of sections, generalizing [SWIII]. We construct classes of examples of these from lifts, twisted lifts and obstructions to lifts through shifted central extensions of groups by the shifted abelian nn-group B n1U(1)\mathbf{B}^{n-1}U(1). Notable examples are String 2-bundles [BaSt] and Fivebrane 6-bundles [SSS2]. The obstructions to lifting ordinary principal bundles to these, hence in particular the obstructions to lifting Spin\mathrm{Spin}-structures to String\mathrm{String}-structures and further to Fivebrane\mathrm{Fivebrane}-structures [SSS2, DHH], are abelian Chern-Simons 3- and 7-bundles with characteristic class the first and second fractional Pontryagin class, whose abelian cocycles have been constructed explicitly by Brylinski and McLaughlin [BML]. We realize their construction as an abelian component of obstruction theory in nonabelian cohomology by \infty-Lie-integrating the L L_\infty-algebraic data in [SSS1]. As a result, even if the lift fails, we obtain twisted String 2- and twisted Fivebrane 6-bundles classified in twisted nonabelian (differential) cohomology and generalizing the twisted bundles appearing in twisted K-theory. We explain the Green-Schwarz mechanism in heterotic string theory in terms of twisted String 2-bundles and its magnetic dual version in terms of twisted Fivebrane 6-bundles. We close by transgressing differential cocycles to mapping spaces, thereby obtaining their volume holonomies, and show that for Chern-Simons cocycles this yields the action functionals for Chern-Simons theory and its higher dimensional generalizations, regarded as extended quantum field theories.

This is work in progress, but maybe already of interest, given the recent discussions here. Comments, especially critical comments, are very welcome and there is room for further collaboration, clearly. There are various aspects here which deserve to be developed further for their own sake.

Here is a quick summary of the global idea from p. 6:


We proceed as follows.

To set ourselves up in a suitably general context of differential geometry we model Spaces{Spaces} as sheaves on CartesianSpaces{CartesianSpaces}, with DiffeologicalSpaces{DiffeologicalSpaces} [BaHo] and SmoothManifolds{SmoothManifolds} contained as subcategories of tame objects.

To have manifest close contact to familiar constructions in homological algebra, differential geometry and physics which we want to reproduce and generalize, the model for \infty-categories which we choose is strict \infty-categories, known as ωCategories\omega{Categories}. This turns out to be not only convenient, admitting all tools of nonabelian algebraic topology [BrHiSi], but also sufficient.

To handle the homotopy theoretic context of ω\omega-categories internal to Spaces{Spaces}, equivalently: ω\omega-category valued sheaves, ωCategories(Spaces)Sheaves(CartesianSpaces,ωCategories), \omega {Categories}({Spaces}) \simeq {Sheaves}({CartesianSpaces}, \omega{Categories}) \,, we obtain from the known homotopy model category structure on ωCategories\omega{Categories} [BrGo, LaMeWo] by stalkwise refinement the structure of a category of fibrant objects [K.-S. Brown]. This yields a homotopy (bi-)category Ho(ωCategories(Spaces))\mathbf{Ho}(\omega{Categories}({Spaces})) whose Hom-spaces realize cohomology in this context, analogous to [Jardine:CocycleCats].

To establish contact with ordinary abelian Čech cohomology with coefficients in complexes of sheaves of abelian groups we consider descent for ω\omega-category valued presheaves [Street] and the corresponding notion of ω\omega-stacks. Dually this leads to a notion of codescent for ω\omega-category valued co-presheaves, which serves to translate from cohomology in terms of descent to cohomology in terms of the homotopy category.

In this context we set up our central definition of twisted differential nonabelian cohomology:

nonabelian cohomology for structure ω\omega-group GG is cohomology with coefficients in hom(Π(),BG)\mathrm{hom}(\Pi(-),\mathbf{B}G), for Π\Pi an ω\omega-category valued copresheaf;

twisted cohomology is a refinement of the obstruction to lifting of cocycles through extensions BG^BG\mathbf{B} \hat G \to \mathbf{B}G

differential cohomology is a refinement of the obstruction to the extension of cocycles along the inclusion of copresheaves 𝒫 0()Π ω()\mathcal{P}_0(-) \hookrightarrow \Pi_\omega(-) of discrete ω\omega-groupoids into fundamental ω\omega-groupoids

As a tool for explicitly constructing (twisted, differential) nonabelian cocycles we describe \infty-Lie theory of smooth ω\omega-groupoids and Lie \infty-algebroids, following [Severa, Getzler, Henriques] as the theory of two consecutive adjunctions relating ωGroupoids(Spaces)\omega{Groupoids}({Spaces}) to Spaces{Spaces} and Spaces{Spaces} to L AlgebroidsL_\infty {Algebroids}. Both adjunctions are examples of Stone dualities induced by ambimorphic objects. The first adjunction is induced by the object of finite paths, Π ω()\Pi_\omega(-), while the second is induced by the object of infinitesimal paths, Ω ()\Omega^\bullet(-)

Using these adjunctions we \infty-Lie integrate the L L_\infty-algebraic cocycles from [SSS1] from L AlgebroidsL_\infty{Algebroids} to ωGroupoids(Spaces)\omega {Groupoids}({Spaces}) to obtain nonabelian differential cocycles when certain integrability conditions are met .

Applied to the String- and Fivebrane- L L_\infty-algebraic cocycles and their Chern-Simons obstructions of [SSS1] this yields an explicit construction of twisted differential cocycles representing twisted String 2- and twisted Fivebrane 6-bundles with connection. The twist itself is the obstruction to obtaining untwisted such bundles and lives in abelian Deligne cohomology where it represents Chern-Simons connections and their higher analogs.

Finally we transgress the differential cocycles thus obtained to mapping spaces and show that the transgressed differential cocycles exhibit the \underline{holonomy} [SWII, SWIII] and can be interpreted as the action functionals for extended Chern-Simons quantum field theories.

Posted at October 30, 2008 7:08 PM UTC

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6 Comments & 4 Trackbacks

Re: Twisted Differential Nonabelian Cohomology

Hi Urs,

That article looks like a very self-contained didatic book, with subjects properly listed on the index and solved exercises at the end. Can I understand it as such?

Posted by: Daniel de Franša MTd2 on October 31, 2008 3:25 AM | Permalink | Reply to this

Re: Twisted Differential Nonabelian Cohomology

That article looks like a very self-contained didatic book, with subjects properly listed on the index and solved exercises at the end.

Just to be very sure that there is no misunderstanding I want to emphasize again that this is work in progress, as one can see from all the internal comments inside the document and from the inclomplete sections.

As far as self-contained goes: for the moment we have that “glossary” section at the end. This arose from discussion who the intended audience of this stuff should be. One important motivation for the entire development is to explain structures and phenomena that arise in quantum field and string theory. But many of the tools we need tend to be (of course there are exceptions) unknown among those who currently appreciate the relavance of the applications.

And conversely. We also have a “glossary” section (very brief at the moment) which goes the other way round and tries to (that’s at least the original intention, in its present state this may well fail at the moment) point out to those familiar with the abstract nonsense that there are interesting applications out there which are waiting to get treated with this abstract stuff.

So that’s where this glossary comes from. It is not the kind of thing that is likely to survive once one thinks about formal publication, but for the time being I think it is good to keep it there. (Of course it would be even better to further expand on it. Volunteers should email me ;-).

and solved exercises at the end.

The second part is titled “Examples and applications” because two things are discussed here, after the development (or what exists of that) of the general theory in the previous part:

i) familiar examples are shown to be reproduced. You might call this “solved exercises”. And ii): the machinery is applied to new examples, in particular to those lifting problems through the sequence of ω\omega-groups Fivebrane(n)String(n)Spin(n)SO(n)O(n)Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to O(n). Well, perhaps this is an just exercise, too. It certainly is in a way in that everything here is supposed to follow straightforwardly by just turning the crank on the machinery developed before.

Posted by: Urs Schreiber on October 31, 2008 5:18 PM | Permalink | Reply to this

Re: Twisted Differential Nonabelian Cohomology

Okay, I am slowly getting up to speed with Urs’s technology. I know it’s not actually Urs’s — it’s basically just a smooth reformulation of rational homotopy theory (not that I understand rational homotopy theory though!) as Urs keeps emphasizing.

Anyhow, Urs can take an L L_\infty-algebra and produce for you a smooth strict ω\omega-groupoid integrating that algebra. (Note: ω\omega-groupoid means strict, strict, strict).

And if you give him a smooth ω\omega-groupoid, he can differentiate it and give you back an L L_\infty-algebra.

John has raised concerns about whether these smooth ω\omega-groupoids are good enough for the long term needs of higher-dimensional algebra. Perhaps one needs to weaken them.

But remarkably, Urs has not come across an application yet that needs anything weaker! His formalism is apparantly capable of constructing all the smooth higher groupoids one needs in Chern-Simons theory. And for now, that’s surely good enough for me.

If you’re a higher-category theorist, and you’re wondering how Urs is getting away with strict stuff while you have been forced to sweat, travail and toil away with BataninTamsamamiStreetPaolisemiweakstrictglobluarsnucategories… well it is a bit of a mystery to me too. I want to get to the bottom of this.

But it’s the word smooth that may well be playing the crucial role. It enables you to formulate everything infinitesimally. And the infinitesimal data of a smooth higher groupoid is an L L_\infty-algebra. Note: these gismos have loads of `weak’ data in them. You want Jacobiators, associators, interchangers, triangulators? They’re all in there.

But when you integrate an L L_\infty-algebra, all that geometric higher coherence data magically gets dissolved into `smoothness’. When you differentiate, out it pops again.

I don’t understand it.

Posted by: Bruce Bartlett on October 31, 2008 12:43 PM | Permalink | Reply to this

Re: Twisted Differential Nonabelian Cohomology

Bruce is wondering # about the context of ω\omega-groupoids (strict \infty-categories with strict inverses).

Here are some remarks:

a) just to make sure, there is not a claim here, at the moment, to the extent “\infty-Lie integration to strict smooth \infty-groupoids is/is not more general/less general in some sense than integration to weak \infty-groupoids”.

What’s happening is that one observes that the Sullivan/Getzler method can be understood as first forming the classifying space of flat L L_\infty-algebra valued forms, and then probing that using a weak path \infty-groupoid. The information about the L L_\infty-algebra is really all in that classifying space, and we just need to see how to extract it. The observation is that for every space we also have its strict (smooth) fundamental \infty-groupoid, and that in the examples we have worked out (in low dimensions mostly, but not only) forming the strict smooth \infty-groupoid of that space yields the strictification of the weak \infty-groupoid.

b) maybe one should keep in mind that L L_\infty-algebras are also not the weakest possible Lie structure: we knew from the get-go (as for instance remarked in John and Alissa’s HDA VI) that there should be weak L L_\infty-algebras which are more general in that they are skew-symmetric only up to higher coherent equivalence. Roytenberg recently worked that out for n=2n=2, as you know.

So maybe a good question to ask is: what would these weak L L_\infty-algebras be like, how would we form their classifying spaces, and how could we probe these? What would weak L L_\infty-algebras integrate to?

You may remember that I made comments on this question a couple of times here:

Let’s ask where the skew-symmetry of a Lie algebra bracket comes from: I think it makes sense to say that it comes from the fact that a Lie group has strict inverses! Because the Lie bracket is the differential of the group commutator ddt| t=0g(t)h(t)g(t) 1h(t) 1\frac{d}{dt}|_{t=0} g(t) h(t) g(t)^{-1} h(t)^{-1} and writing this out with exponential maps and looking at the familiar formulas from a higher perspective, one sees that the skew symmetry of the bracket reflects the fact that g(t) 1g(t)^{-1} really is the inverse of g(t)g(t), not up to something.

So, I am thinking that: maybe it makes sense to integrate L L_\infty-algebras to strict \infty-groupoids, because weak \infty-groupoids should be the integrated versions of weak L L_\infty-algebras!

This is just an idea, I don’t have any precise statement here. But maybe it is an idea worth of keeping in the present context.

c) Or maybe not. Maybe forming the strict path \infty-groupoid of the classifying space of flat L L_\infty-algebras does lose information. As I mentioned elsewhere, I currently don’t have a complete idea (though some partial ideas) of what should be the “third \infty-Lie theorem”. Instead, ω\omega-categories and ω\omega-groupoids currently serve for me the purpose that they are tractable and that we can use them in practice to attack the questions that ought to be attacked. I keep in mind the option of eventually weakening the entire setup and allowing weak \infty-categories. But for the moment sticking with the strict ones allows to avoid getting lost in just foundational and technical questions and to instead make some progress on applications.

Sjoerd Crans somewhere in his articles says a wise word along the lines (just roughly paraphrasing from memory): before trying to tackle weak \infty-categories let’s make sure we have completely mastered the strict ones.

He is speaking of the general theory. But the same kind of comment should apply to applications of \infty-categories: let’s see how many applications can be handled already with strict \infty-categories. We can still eventually try to weaken everyhting if we see we are getting stuck otherwise.

Posted by: Urs Schreiber on October 31, 2008 2:21 PM | Permalink | Reply to this

Re: Twisted Differential Nonabelian Cohomology

Ok, it’s starting to come back to me a little bit. I used to know some of these things, vaguely, but I forgot.

Two questions I’ve had recently:

(a) When you form the smooth space K(G)K(G) corresponding to a smooth ω\omega-groupoid GG, how come it isn’t just the smooth space of 0-cells, ie. K(G)=G 0K(G) = G_0, as opposed to the construction K(G)=Hom(Π ω(),)K(G) = Hom(\Pi_\omega(\cdot), \cdot). I guess my construction is way off, because for a group it would just give you a smooth space consisting of a single point. But it shows I have a bit of confusion here.

(b) Have you shown that integrating an L L_\infty-algebra to a smooth ω\omega-groupoid and then differentiating it again indeed reproduces the original L L_\infty-algebra? I ask because I couldn’t quite find this point explicitly dealt with in Andre Henriques’ paper either. Perhaps I have a misconception.

(c) Also, you have broken down the process of differentating smooth ω\omega-groupoids into L L_\infty-algebras into a two step process. Given a smooth ω\omega-groupoid GG, first you form the geometric model (what is the word for this) K(G)K(G), then you convert this into an L L_\infty-algebra. You do it by some clever abstract nonsense involving hom’s into and out of tautological things.

But one might also imagine a more pedestrian approach, where given a smooth ω\omega-groupoid GG you obtain an L L_\infty algebra 𝔤\mathfrak{g} by literally differentiating it, in the good old way:

(1)ddt| t=0curveofnmorphisms. \frac{d}{dt}|_{t=0} curve of n-morphisms.

Does this more pedestrian approach actually make sense, and can you show that this is the same thing (up to weak equivalence)?

I’d like to be able to tell people: “To get an L L_\infty-algebra from a smooth groupoid, we just take the tangent space of the 0-cells, 1-cells, 2-cells etc. at the identity”. No abstract nonsense :-) (though there is nothing wrong with that!). Do these concepts like “tangent space” work out nicely for smooth spaces?

Posted by: Bruce Bartlett on October 31, 2008 6:27 PM | Permalink | Reply to this

Re: Twisted Differential Nonabelian Cohomology

(a) When you form the smooth space K(G)K(G) corresponding to a smooth ω\omega-groupoid GG, how come it isn’t just the smooth space of 0-cells, ie. K(G)=G 0K(G) = G_0, as opposed to the construction K(G)=Hom(Π ω(,)G)K(G) = Hom(\Pi_\omega(-,)G). I guess my construction is way off, because for a group it would just give you a smooth space consisting of a single point. But it shows I have a bit of confusion here.

Here I am not sure what the question really is! As you say, one needs to “probe” all higher cells of GG.

More abstractly, this is what happens to be the right adjoint to Π ω:SpacesωCategories(Spaces)\Pi_\omega : Spaces \to \omega Categories(Spaces). See the proof of prop. 4.16, p. 66.

(b) Have you shown that integrating an L L_\infty-algebra to a smooth ω\omega-groupoid and then differentiating it again indeed reproduces the original L L_\infty-algebra?

Only for special cases in low dimensions so far.

It turns out that there, up to the case of strict Lie 2-algebras and Lie 2-groups, this is really just a direct corollary, just a rereading of the theorem in [SWII], which says that Hom(P 2(U),BG)=Ω (U,g) Hom(P_2(U), \mathbf{B}G) = \Omega^\bullet(U,g) for GG a Lie 2-group and gg its strict Lie 2-algebra, which implies for the smooth fundamental 2-groupoid Hom(Π 2(U),BG)=Ω flat (U,g). Hom(\Pi_2(U), \mathbf{B}G) = \Omega^\bullet_{flat}(U,g) \,. So this says that as sheaves we have Hom(Pi 2(),BG)=S(CE(g)) Hom(Pi_2(-),\mathbf{B}G) = S(CE(g)) !

This is currently mentioned, a bit briefly in section 5.3, p. 89.

But what I don’t have, and what nobody had last time I checked (but let’s see what Pavol &Scaon;evera will say in his talk in Lausanne next week!) is a “third \infty-Lie theorem” which tells us what exactly happens when we first integrate and then differentiate.

I was thinking it is a good first step to realize that the integration and differentiation procedure is the concatenation of two adjunctions. I regard that as a first, albeit incomplete, step at “third \infty-Lie”.

Notice that in Sullivan models in rational homotopy theory the crucial theorem which makes everything work is the one which says that the unit of the adjunction

Algebras \to Spaces \to Algebras

is, while not an isomorphism, a weak isomorphism, namely an isomorphism on cohomology!

So this means that when you start with a (nice) DGCA AA (a Sullivan algebra), then build a space S(A)S(A) from that, and then take the differential forms on that space, Ω (S(A))\Omega^\bullet(S(A)), you’ll have a DGCA much larger than the original DGCA AA.

But there is a canonical injection AΩ (S(A)) A \hookrightarrow \Omega^\bullet(S(A)) (which is the unit of the adjunction) and in rational homotopy theory one proves that this injection is a weak isomorphism an isomorphism in cohomology.

This is theorem 1.24 1), p. 10 in Kathryn Hess’s review.

(This works for Sullivan algebras, which are the cofibrant objects in DGCAs, as we discussed by email, last big paragraph on p. 6).

I ask because I couldn’t quite find this point explicitly dealt with in André Henriques’ paper either. Perhaps I have a misconception.

Last time I checked, which is a few months ago, it was not known to which degree precisely Ševera’s differentiation is inverse (left or right) to Getzler’s integration.

As I say, maybe that has changed by know. We’ll listen to what Pavol has to say in Lausanne!

Does this more pedestrian approach actually make sense, and can you show that this is the same thing (up to weak equivalence)?

I think heuristically this is precisely what the abstract nonsesense proceudure does for you.

Maybe have a look at section 5.2.1, p. 77, where I spell out how precisely the abstract nonsense of homming around reproduces exactly the constructions that Crainic and fernandes review in their review Integrability of Lie brackets.

Do these concepts like “tangent space” work out nicely for smooth spaces?

Well, the more immediate natural concept is cotangent space, which constitute indeed one part of the adjunctions, namely the object of infinitesima paths Ω :Spaces opDGCAs. \Omega^\bullet : Spaces^{op} \to DGCAs \,.

Notice that the deRham DGCA Ω (X)\Omega^\bullet(X) of a smooth space XX is the Chevalley-Eilenberg algebra of its tangent Lie algebroid Ω (X)=CE(TX). \Omega^\bullet(X) = CE(T X) \,.

Posted by: Urs Schreiber on October 31, 2008 7:09 PM | Permalink | Reply to this
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