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July 17, 2008

News on Measures on Groupoids?

Posted by Urs Schreiber

Over at his Theoretical Atlas blog, Jeffrey Morton reports from Quantum Gravity and Quantum Geometry 2008, briefly indicating the content of a couple of talks. This one here especially caught my attention:

Benjamin Bahr gave another talk dealing with categorical issues - namely, how to get measures on certain groupoids, such as, indeed, the groupoid of connections on a manifold. In fact, he treated various cases under the same framework: flat and non-flat connections, on manifolds and on graphs - and others.

I have posted the following comment, which however is still “awaiting moderation”. While it awaits, I thought I’d share this here:

Hi,

thanks for the report!

I am thrilled by this bit:

Benjamin Bahr gave another talk dealing with categorical issues - namely, how to get measures on certain groupoids, such as, indeed, the groupoid of connections on a manifold.

I had not heard of Bahr’s work before. A quick search on the web didn’t yield anything (but maybe I didn’t search carefully enough). But I am very interested in this general question. Have thought a bit about it myself.

Do you happen to have any further details on this? Is there anything written available in any form?

Do you know if his groupoid measures are at all related to Tom Leinster’s weightings on categories I, II, III, IV?

Does Bahr go as far as saying something about path integrals for gauge theory? Does he mention connections to BRST-BV methods which are the standard tool for handling integration over things like the groupoid of gauge connections?

For the case of finite groups and hence necessarily flat connections, the groupoids they live in are, naturally, well understood. More strikingly, it is known in these cases that the right “path integral” (just a sum in this case) measure on these groupoids – which appears in Dijkgraaf-Witten theory but also in its higher version, the Yetter-Martins-Porter model that you mention above – is precisely the Baez-Dolan-Leinster groupoid measure for Dijkgraaf-Witten, and its higher version for the Cat-enriched case for the Martins-Porter model. (I recall this for DW and prove it for 2DW in section 1.4 of On Σ\Sigma-models and nonabelian differential cohomology).

Does Bahr’s approach say anything about this? Does he reproduce the measures for DW and higher DW theory? Do you happen to know?

I’d be grateful for whatever information you might have.

Posted at July 17, 2008 9:41 AM UTC

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16 Comments & 1 Trackback

Re: News on Measures on Groupoids?

His abstract is a start.

On the quantization of connection theories with categories

The way one quantizes the kinematical structure of Loop Quantum Gravity is deeply connected to category theory. Fundamental is the notion of the path groupoid, i.e. the collection of all paths in a manifold. In this language e.g. connections, gauge transformations and diffeomorphisms arise as functors, natural transformations and (path groupoid) automorphisms.

In this talk it will be shown how similar constructions can be carried out for any sufficiently well-behaved groupoid, and that the choice of groupoid, which only contains combinatorial information about paths, already determines analytical properties such as a topology and a measure (the analogue of the Ashtekar-Isham-Lewandowski measure) on quantum configuration space. Both are automatically invariant under the action of gauge transformations and groupoid automorphisms (which play the role of “diffeomorphisms” in this context). The uniqueness of measure and topology will be discussed.

Examples for these structures are, besides LQG, also lattice gauge theory and Chern-Simons theory.

Posted by: David Corfield on July 17, 2008 10:11 AM | Permalink | Reply to this

Re: News on Measures on Groupoids?

Ah, thanks!

Hm, if it is the Ashtekar-Lewandowski measure he is trying to generalize then I see the usual problems ahead.

The abstrac sounds as if he is just observing that looking at functors from a groupoid of paths to somewhere is not much different from looking at functors from any groupoid to that somewhere (if one ignores all exra structure, such as topology and smoothness).

In any case, he is apparently not doing anything along the lines of the Baez-Dolan-Leinster groupoid measure.

Posted by: Urs Schreiber on July 17, 2008 10:23 AM | Permalink | Reply to this

Re: News on Measures on Groupoids?

I am wondering if anyone working on these generalized connections (= parallel transport functors that are allowed to ignore the smooth and the topological structure of the space of paths) has noticed Konrad’s and my theorem 3.12 which characterizes the smooth transport functors coming from possibly nontrivial smooth principal GG-bundles among the “generalized connections” as precisely those that have smooth Wilson lines.

Posted by: Urs Schreiber on July 17, 2008 10:32 AM | Permalink | Reply to this

Re: News on Measures on Groupoids?

It is actually possible to find a complete list of the slides (and some of videos) of the talks presented at the QG2-2008 conference in Nottingham at the following web-page:

http://echo.maths.nottingham.ac.uk/qg/wiki/index.php/QGsquared-slides

The slides of Benjamin Bahr’s talk (Monday afternoon, parallel sessions) are here.

Best Regards.

Posted by: Paolo Bertozzini on July 18, 2008 10:44 AM | Permalink | Reply to this

Re: News on Measures on Groupoids?

Thanks.

Here are some random comments on these slides, to whoever it may concern.

Slide 45 defines

Aut loc.smallGrpds(C) Aut_{loc. small Grpds}(C)

to “act locally” if some property holds.

I am not sure which examples of locally small groupoids are claimed to have this property.

For instance, what is the groupoid structure for the second and third example on p. 48?

This is related to the discussion of thin-ness above. I am not sure I understand what the “piecewise analytic” and “piecewise smooth” is supposed to be good for: since parallel transport of a connection is invariant under reparameterization and zig-zags (thin homotopy) anyway a path can look very “kinky” in target space while still being a smooth map from the interval to target space: that map just has to have all derivatives vanishing at the kink.

So this is related to this general problem I have: what is denoted A¯\bar A in these slides (def. on slide 40) is actually not a space of connections in the usual sense, but a space of “generalized connections”. This doesn’t seem to be mentioned in these slides.

I think this has for instance the consequence that the Hilbert space mentioned as point 2 on slide 53 is not a separable Hilbert space. If so, that deserves to be mentioned, since quantization of gauge theory is usually demanded to give a separable Hilbert space.

Similarly the statement about the uniqueness of the measure on the “space of connections”: for U(1)U(1)-gauge theory the space of connections, its measure as well as the (separable) Hilbert space corresponding to it can be and has been defined – and neither ingredient coincides with what is discussed in these slides.

From the title I had thought this was a new approach, but it seems it is just the familiar prescription used in LQG with the word functor now made explicit where it had been implicit mostly all along. No?

Posted by: Urs Schreiber on July 18, 2008 12:33 PM | Permalink | Reply to this

Re: News on Measures on Groupoids?

What is the definition of “path” in this context?

Thank you.

Posted by: Christine Dantas on July 17, 2008 12:29 PM | Permalink | Reply to this

Re: News on Measures on Groupoids?

What is the definition of “path” in this context?

A path in a space XX in this context is an equivalence class of smooth maps [0,1]→X [0,1] \to X with the property that they are constant in a neighbourhood of the boundary of the interval (have “sitting instants”), where

- two such maps define the same path if they differ by reparameterization and zig-zag-moves.

More precisely, two such maps define the same path if there is a smooth homotopy Σ:[0,1] 2→X \Sigma : [0,1]^2 \to X between them which is “thin”. Σ\Sigma being “thin” means that the rank of the differential

Σ *:T σ[0,1] 2→T Σ(σ)X \Sigma_* : T_\sigma[0,1]^2 \to T_{\Sigma(\sigma)}X

is everywhere smaller than 2.

In other words, paths are the quotients of parameterized maps from the interval under the minimal equivalence relation which makes concatenation of paths along their endpoints the composition operation in a smooth groupoid.

Posted by: Urs Schreiber on July 17, 2008 7:37 PM | Permalink | Reply to this

Re: News on Measures on Groupoids?

Thank you for a precise and fast response. The “thin” jargon is new to me.

Kind regards.

Posted by: Christine Dantas on July 17, 2008 7:49 PM | Permalink | Reply to this

Re: News on Measures on Groupoids?

The “thin” jargon is new to me.

You probably know it from gauge theory under the term “zig-zag symmetry” or “zig-zag invariance” of Wilson loops. For instance here.

The “thin homotopy” way of looking at it has the advantage

a) that it generalizes more obviously to higher dimensional paths

b) that it explicitly states the mathematical condition which you actually need to invoke in proofs, such as the (standard) proof that Wilson loops are independent of the homotopy class of the loop precisely if the connection is flat, etc.

Finally, I should add that I think the best and most robust formulation of the condition is actually dual to the standard one I gave: a map

V:[0,1] n→XV : [0,1]^n \to X

is thin precisely if all nn-forms pulled back along this map vanish, i.e. if

V *:Ω n(X)→Ω n([0,1] n) V^* : \Omega^n(X) \to \Omega^n([0,1]^n)

is the zero map. In this form, the thin-ness condition generalizes from manifolds to more general smooth spaces, such as loop spaces, diffeological spaces etc.

This way of fomulating it was introduced here.

Posted by: Urs Schreiber on July 18, 2008 10:37 AM | Permalink | Reply to this

Re: News on Measures on Groupoids?

Thank you very much for the further comments and the indicated references.

I suppose one could use the same thin-ness condition in spaces where a local partial order is defined, so that you have paths with a “direction”?

Thanks.

Posted by: Christine Dantas on July 18, 2008 2:57 PM | Permalink | Reply to this

Re: News on Measures on Groupoids?

I had a very nice conversation with Alan Weinstein yesterday. He explained how he has generalized ‘groupoid cardinality’ from discrete groupoids to Lie groupoids (or differentiable stacks). Not only does he get a generalization of cardinality, or ‘volume’, for differentiable stacks, he also studies a kind of ‘measure’ on such things, whose integral, when finite, gives the volume. I hope to explain this soon in This Week’s Finds.

Posted by: John Baez on July 18, 2008 9:20 AM | Permalink | Reply to this

Re: News on Measures on Groupoids?

I had a very nice conversation with Alan Weinstein yesterday. He explained how he has generalized ‘groupoid cardinality’ from discrete groupoids to Lie groupoids

That’s fantastic. I am looking forward to seeing the details.

Posted by: Urs Schreiber on July 18, 2008 9:54 AM | Permalink | Reply to this

Re: News on Measures on Groupoids?

Before you get too excited, I’ll warn you right away: the measure on a Lie groupoid (or differentiable stack) is not completely god-given, as it was in the topologically discrete case that Jim Dolan and I studied.

For example, when the Lie groupoid has an nn-dimensional manifold of objects (n>0n \gt 0) and only identity morphisms, any volume form on this manifold gives an equally acceptable measure.

But in many cases, Weinstein’s definitions restrict the choice of acceptable measure quite a lot. So, it’s an interesting nontrivial idea.

Posted by: John Baez on July 18, 2008 1:45 PM | Permalink | Reply to this

Re: News on Measures on Groupoids?

the measure on a Lie groupoid (or differentiable stack) is not completely god-given

Okay, I can accept that.

But maybe if we just add a little bit of extra structure… such as … having a symplectic groupoid … and a solution to some “master equation”…

Posted by: Urs Schreiber on July 18, 2008 2:55 PM | Permalink | Reply to this

Re: News on Measures on Groupoids?

I’ve put the “slides” from my talk at Poisson 2008 (EPFL Lausanne) on volumes of differentiable stacks on my web page (click on my name below) under the heading – Lectures.

You’ll see there exactly what the extra data is which is necessary for the definition of a “measure” on the stack presented by a proper groupoid. Even in the case of a symplectic groupoid, this extra data is necessary. The only case where it seems to be canonical is that of a group acting on itself or on its Lie algebra.

I’m working on a manuscript about all this and hope to have it out before too long.

Posted by: Alan Weinstein on July 19, 2008 10:21 PM | Permalink | Reply to this

Re: News on Measures on Groupoids?

Alan Weinstein,

Modular classes and the volume of a differentiable stack (“slides” from lecture at Poisson 2008, EPFL Lausanne, July 7, 2008)

Thanks! I’ll have a look.

Posted by: Urs Schreiber on July 21, 2008 8:27 PM | Permalink | Reply to this
Read the post The Volume of a Differentiable Stack
Weblog: The n-Category Café
Excerpt: Weinstein's generalisation of the Baez-Dolan groupoid cardinality concept to stacks.
Tracked: September 15, 2008 3:44 PM

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