Higher and Graded Geometric Structures in Göttingen

November 19, 2008

Higher and Graded Geometric Structures in Göttingen

Posted by Urs Schreiber

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There is a mini-workshop extending the Born-Hilbert seminar in Göttingen, organized by Chenchang Zhu.

CRCG Miniworkshop
Higher and graded geometric structures
Monday, Nov. 24
room: “Sitzungszimmer” of the Mathematical Institute Göttingen.

The schedule is:

Ping Xu (Penn State)
Groupoid extensions and Non-ablian gerbes
9:30-10:30

Matthieu Stiénon (Penn State)
2-Group bundles and characteristic classes
11:00-12:00

Christian Blohmann (Regensburg)
Groupoid symmetry of general relativity
13:15-14:15

Rajan Metha (Washington University)
On models for the adjoint representation of a Lie algebroid
14:15-15:15

Urs Schreiber (Hamburg)
Differential non-abelian cohomology
16:00-17:00

Posted at November 19, 2008 10:41 AM UTC

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11 Comments & 3 Trackbacks

updated info

Dear Urs,

The title “supermanifolds” was only a tentative one, based on what Chenchang thought I might speak about. The actual title of my talk will be “On models for the adjoint representation of a Lie algebroid”, and it will be based on this paper with Alfonso Gracia-Saz. But, to be sure, some “super” stuff will make an appearance.

I look forward to meeting you. I’m sure we’ll have a lot to talk about.

Posted by: Rajan Mehta on November 19, 2008 2:38 PM | Permalink | Reply to this

Re: updated info

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Dear Rajan,

thanks for update! I have incorporated it into the above entry now.

By the way, concerning adjoint reps of Lie algebroids: I haven’t had a chance yet to look at your article, but it reminds me that a few days ago in Lausanne we had a talk by Arias Abad on his thesis which involved adjoints reps of a Lie-groupoid on its Lie algebroid. Do you happen to be aware of that?

Another by the way: I had first (conciously) come across your name and your work last summer when I was in Bonn. (I had mentioned that here). Back then I was being pointed to and was looking at your Supergroupoids, double structures and equivariant cohomology in order to compare notions, because with Christoph Sachse we were thinking about $\infty$ -Lie integrating the supergravity Lie 3-algebra $sugra_{\mu_4}(10,1)$ (which is an “String-like” extension # of the super-Poincaré Lie algebra by $b^2 u(1)$ using a 4-cocycle $\mu_4$ ) to a super Lie 3-group, i.e. a 3-group internal to supermanifolds.

We saw two options for how to generalize $\infty$ -Lie theory from ordinary to super-spaces:

a) define the super-path fundamental super- $\infty$ -groupoid $\Pi^{1|N}(X)$ of a smooth superspace $X$ and then compute $\Pi^{1|N}(S(CE(sugra_{\mu_4}(10,1))))$ (where $S(CE(sugra_{\mu_4}(10,1)))$ is the smooth super-classifying space of flat $sugra_{\mu_4}(10,1)$ -valued superforms)

b) consider the super $L_\infty$ -algebra $sugra_{\mu_4}(10,1)$ as a presheaf $sugra_{\mu_4}(10,1) : Superpoints^{op} \to L_\infty$ on superpoints with values in ordinary $L_\infty$ -algebras, integrate this super-pointwise using the ordinary bosonic theory to get the presheaf $\mathbb{R}^{0|n} \mapsto \Pi_\omega( S(CE(sugra_{\mu_4}(10,1)(\mathbb{R}^{0|n}))) )$ and interpret that smooth $\infty$ -groupoid valued presheaf on superpoints in turn as a $\infty$ -groupoid internal to smooth superspaces.

We started with a) but quickly ran into calculational problems with that. Christoph Sachse then worked hard on carrying through a) for simpler toy examples such as the super-Lie 1-algebra $\mathbb{R}^{1|1}$ with the nontrivial $[\theta,\theta]$ -bracket, and even that led to surprisingly subtle computational issues with those super-homotopies of superpaths.

So in the end I decided that if there is any progress to be obtained here, then along b). But I don’t know, maybe we were not being clever enough.

I am just saying all this in case it rings a bell with anything you have been thinking about. I’d be interested in discussing this. The super-Lie 3-group integrating $sugra_{\mu_4}(10,1)$ is expected to be very interesting, analogous to the $String(n)$ -2-group (and in fact appearing together with that, in some way, as the structure $n$ -group of the 2-gerbe appearing in 11-d quantum supergravity (aka “M-theory”)).

Posted by: Urs Schreiber on November 19, 2008 5:04 PM | Permalink | Reply to this

Re: updated info

I knew that Camilo has been working on representations up to homotopy, but I haven’t yet had the chance to find out what his thesis is about. I found the abstract for his talk, and it sounds interesting. If we are trying to interpret VB-algebroids as Lie algebroid representations, then the next logical step would be to consider the “integrated” picture, where VB-groupoids are interpreted as Lie groupoid representations. At this stage, we should find some relations with the adjoint rep up to homotopy of a Lie groupoid.

Regarding the integration problem: In my thesis I actually avoided dealing with the issue of integrating Lie superalgebroids to Lie supergroupoids, precisely because I didn’t want to mess around with those technical issues you mentioned. But perhaps in Gottingen we can attempt some of those toy examples and see if we can make some progress.

Posted by: Rajan Mehta on November 19, 2008 8:20 PM | Permalink | Reply to this

Re: updated info

Urs has given us several papers to consider.
Which or other would use the language of:
representations up to homotopy

Posted by: jim stasheff on November 21, 2008 1:34 PM | Permalink | Reply to this

Re: updated info

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Which or other would use the language of: representations up to homotopy

To my shame I have to admit that I still haven’t looked at Rajan Mehta’s On models for the adjoint representation of a Lie algebroid .

But I was wondering, too:

it seems to me that a good definition of reps of $L_\infty$ -algebroids $g$ with CE-algebra $CE(g)$ on comoplexes of vector spaces $V$ is as extensions $\wedge V \leftarrow CE_\rho(g,V) \leftarrow CE(g)$ which is supposed to be dual to the corresponding sequences of an action $\infty$ -groupoid $V \to V//G \to \mathbf{B}G \,.$

We saw that, reformulated equivalently, this is what has been considered in the literature under different names, for instance by Jonathan Block in terms of “superconnections”, described here.

Then, I think for every $L_\infty$ -algebra $g$ there is canonically the adjoint action of $g$ on itself, where $CE_{\rho_{Ad}}(g,g)$ is built much like the Weil algebra $W(g)$ , but with one copy of $g$ shifted down instead of up.

So therefore I haven’t thought about adjoint reps of $L_\infty$ -algebroids with space of objects bigger than the point, since that would appear to be problematic with respect to shifting down.

There are other ways to look at this: the adjoint action of an $n$ -group on itself is given by its inner automorphism $(n+1)$ -group # which one can define for any $n$ -groupoid, too. Differentiating that should yield the adjoint rep of the $n$ -groupoid on its Lie $n$ -algebroid.

I am not sure I can see how to get from there to a definition of an action of the Lie $n$ -algebroid on itself. Probably somehow by differentiating once again or so.

Well, I should just look at Rajan’s article, I suppose. :-)

Posted by: Urs Schreiber on November 21, 2008 6:08 PM | Permalink | Reply to this

Re: updated info

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I should have been more precise: the $V$ I mentioned is in general of course a complex of vector bundles over the space of objects of the $L_\infty$ -algebroid acting on it.

Posted by: Urs Schreiber on November 21, 2008 6:18 PM | Permalink | Reply to this

Re: updated info

Hmm, I should look at Block’s paper, since superconnections played a big role in our paper also. But, more fundamentally, we did have a Lie algebroid extension $$V \to \hat{A} \to A$$ which looks like the dual of the extension of complexes that you wrote, except that in our case the $V$ isn’t the complex of vector bundles that $A$ acts on, but rather the bundle of degree -1 endomorphisms of said complex.

It seems like this could be generalized to $L\infty$-algebroids without too much difficulty. At the very least, the notion of an $L\infty$-algebroid object in the category of vector bundles is a sensible one.

hmm, I’m not sure if I’m embedding the latex correctly in my comment, so sorry if it doesn’t come out looking right…

Posted by: Rajan Mehta on November 22, 2008 5:29 PM | Permalink | Reply to this

Re: updated info

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You need to choose the right text filter from the menu (probably “itex to MathML with parbreaks”—that’s the one I always use). You can tell how it’s going to turn out from the preview.

Posted by: Tim Silverman on November 22, 2008 5:53 PM | Permalink | Reply to this

Re: updated info

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Yes, it’s probably closely related or even the same notion. Good.

I am in a bit of a hurry now. Let’s talk about this tomorrow.

Posted by: Urs Schreiber on November 23, 2008 11:51 AM | Permalink | Reply to this

Re: Higher and Graded Geometric Structures in Göttingen

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I have prepared “background” notes for my talk today:

Differential nonabelian cohomology:
Characteristic classes and -forms of $\mathrm{String}(n)$ -principal 2-bundles
(pdf, 6 pages + references)

Abstract. Nonabelian cohomology generalizes Čech cohomology with coefficients in sheaves of complexes of abelian groups to cohomology with coefficients in sheaves of $\infty$ -categories. It classifies higher principal bundles and their higher gerbes of sections. There is a differential refinement which classifies higher bundles with connection.

Interesting examples arise from lifts, and obstructions to lifts, of structure groups through shifted abelian extensions, notably through the $\infty$ -categorical Whitehead tower $\mathrm{Fivebrane}(n) \to \mathrm{String}(n) \to \mathrm{Spin}(n) \to \mathrm{SO}(n) \to \mathrm{O}(n)$ of the group $\mathrm{O}(n)$ . As an application we discuss $\mathrm{String}(n)$ -principal 2-bundles with connection, their characteristic classes and characteristic forms.

The first part on the general setup is the same as in my talk at Higher structures 2008 #, the second part, the one about characteristic classes and forms, deals with material appearing in section 3.4, p. 62 (theory) and 5.5, p. 103 (first examples) of Twisted differential nonabelian cohomlogy #.

This is supposed to make contact with the other talks today, notably with Mathieu Stiénon’s talk on 2-bundles and characteristic classes.

Posted by: Urs Schreiber on November 24, 2008 9:22 AM | Permalink | Reply to this

Re: Higher and Graded Geometric Structures in Göttingen

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Here are notes taken in Christian Blohmann’s talk which is just over

Groupoid symmetry of general relativity (pdf notes)

I didn’t have time yet to go through these again and polish them, so this is a bit like a raw stream.

Consider canonical general relativity by choosing of a spatial (Cauchy) hypersurface and then do the canonical analysis for time evolution from there. As is well known, the constraint algebra (the algebra of symmetries on the phase space which one would want to “quotient by” somehow) is not a Lie algebra: this is often put by physicists by saying that it does not have structure constants but “structure functions” or that “the constraint algebra does not close”.

It seems to me that in one sentence, Christian’s message now is this:

The phase space of general relativity is not a global quotient, but a more general Lie groupoid which is not the action groupoid of a group acting on a space. Accordingly, the constraint algebra is that of sections of the corresponding Lie algebroid.

My question after the talk was: people doing BV-BRST must have known this somehow? But apparently not (?).

So this is an interesting result, it seems. I hope I am stating it correctly, if you are dubious, better have a look at the notes and try to interpolate from this faint shadow of the talk what Christian Blohmann has been doing in, I should mention, joint work with Alan Weinstein.

Posted by: Urs Schreiber on November 24, 2008 3:15 PM | Permalink | Reply to this

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November 19, 2008