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April 19, 2007

Some Conferences

Posted by Urs Schreiber

Busy with reducing the write-up lag. No time to blog.

I am trying not to try to go to too many events, but here are two more I might not be able to resist trying to attend (which may still fail even if I try).

This beautiful one

Principal Bundles, Gerbes and Stacks

17-23 June, 2007 - Bad Honnef, Germany

is right around the corner for me. At least Konrad Waldorf will go and talk about our stuff.

Not sure if John has mentioned this one anywhere yet except on his lectures site, but even if so it’s well worth mentioning it twice:

The Abel Symposium 2007

August 5 - 10 2007, Oslo, Norway

John will talk about

Higher Gauge Theory and Elliptic Cohomology

The concept of elliptic object suggests a relation between elliptic cohomology and “higher gauge theory”, a generalization of gauge theory describing the parallel transport of strings. In higher gauge theory, we categorify familiar notions from gauge theory and consider “principal 2-bundles” with a given “structure 2-group”. These are a slight generalization of nonabelian gerbes. After a quick introduction to these ideas, we focus on the 2-group StringG\mathrm{String}G associated to any compact simple Lie group GG. We describe how this 2-group is built using a central extension of the loop group ΩG\Omega G, and how the classifying space for StringG\mathrm{String}G-2-bundles is related to the “string group” of elliptic cohomology. If there is time, we shall also describe a vector 2-bundle canonically associated to any principal 2-bundle, and how this relates to the von Neumann algebra construction of Stolz and Teichner.

If it were not for the write-up lag, I could point to a concrete paper concerning the last part. But today I wrote at least this informal summary of what I can say about this issue so far:

On 2-Representations and 2-Vector Bundles

There is a canonical 2-representation ρ\rho of any strict 2-group G 2G_2 on Bim2Vect \mathrm{Bim} \subset 2\mathrm{Vect}. This allows to consider 2-vector transport 2-functors tra:P 2(X)2Vect\mathrm{tra} : P_2(X) \to 2\mathrm{Vect} which is locally ρ\rho-trivializable. This gives a notion of ρ\rho-associated 2-vector bundles with connection.

For the simple case G 2=ΣU(1)G_2 = \Sigma U(1) these 2-functors have local semi-trivializations which are line bundle gerbes with connection.

It seems that everything goes through entirely analogously for the String 2-group G 2=String k(G)G_2 = \mathrm{String}_k(G). We naturally obtain a notion of String bundle with 2-connection this way. The resulting 2-functor is superficially different from the String connection 2-functor proposed by Stolz and Teichner, but both share a couple of striking similarities.

It was a while after we found that the big String-1-group is just the realization of the nerve of an infinite-dimensional Lie 2-group that I began wondering if the rep of the String-1-group on von Neumann algebras considered by Stolz-Teichner can be understood in fact as a 2-representation of the corresponding 2-group.

It turns out that there is a simple construction which allows one to canonically represent every strict 2-group on the 2-category of bimodules. I thought about this while at the Weizmann institute in Tel Aviv last year, and reported about it in

Remarks on 2-Reps

At this point the construction was done just at the level of groups in sets, i.e. without taking into account any further structure. For applying this idea to the String 2-group one needs to properly deal with highest weight representations of loop groups, the von Neumann algebras generated from these and the Connes fusion tensor product of the corresponding bimodules.

I reported on observations how to make this work in

Remarks on String(n).

I am not an expert on the functional analysis needed here, but meanwhile two experts I met indicated, after I had explained the idea, that indeed the construction of the canonical 2-rep should go through smoothly also in Bim vN\mathrm{Bim}_{\mathrm{vN}}.

If this is indeed the case (and it seems that all is missing is a clean write-up of the proof), then we indeed get a neat way to talk about String connections.

Since this would be then just a special case of the general theory of associated 2-transport it pays to warm up by playing this through in full detail for the canonical rep of the 2-group ΣU(1)\Sigma U(1) on Bim\mathrm{Bim}. This leads to rank-1 2-vector bundles which are equivalent to bundle gerbes:

On n-Transport: 2-Vector Transport and Line Bundle Gerbes

The point is that the String case works almost entirely analogously, also due to the fact that in both cases the realization of the nerve of the structure 2-group acts as the automorphism 2-group of the algebra that the canonical 2-representation is built from

2-Groups and Algebras.

By turning the crank, we hence find that the 2-vector bundle canonically associated to a String 2-bundle “is” a bundle of von Neumann algebras (just like a line bundle gerbe “is” a bundle of compact operators!) – and that an associated 2-transport in that bundle is indeed a 2-functor from 2-paths to von Neumann bimodules, coinciding on points with the fibers of that algebra bundle.

This result looks strikingly similar to the 2-functorial definition of String connections that Stolz and Teichner had proposed. It would be strange if it weren’t essentially the same concept. But for the moment I haven’t managed to give a direct translation from asscoiated String 2-transport to the Stolz-Teichner definition in detail.

See the end of the above pdf for more on that.

Posted at April 19, 2007 7:41 PM UTC

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4 Comments & 4 Trackbacks

Mlodinow; Re: Some Conferences

A von Neumann algebra is a strongly closed *-subalgebra of the algebra B(H) of bounded operators on a Hilbert space H, or, by double commutant theorem, it is a *-subalgebra A of B(H) that is equal to its double commutant.

I am interested in *-algebras, and the fact that von Neumann was trying to generalize the foundations of Quantum Mechanics.

Does that suggest that the 2-vector bundle canonically associated to a String 2-bundle can be projected to a representation of Quantum Mechanics, but also to alternative quasi-QM systems?

I recall that Gell-Mann brought Leonard Mlodinow to Caltech because of his PhD dissertation on how QM would work if there were an infinite number of spacial dimensions.

Posted by: Jonathan Vos Post on April 21, 2007 4:21 PM | Permalink | Reply to this

Re: Mlodinow; Re: Some Conferences

Does that suggest that the 2-vector bundle canonically associated to a String 2-bundle can be projected to a representation of Quantum Mechanics, but also to alternative quasi-QM systems?

It is true that one encounters the curious phenomenon that upon categorifying things related to vector bundles one runs into lots of structure that has a quantum-mechanical smell to it, without any quantum mechnical systems having manifestly entered the picture.

I cannot say that I fully understand what this is telling us, if anything. But there are a couple of aspects of it which are understood.

In his TWFs, John has emphasized this point for a long while, mostly in the context of the iterated classifying spaces of the integers, where one gets U(1)B() U(1) \simeq B(\mathbb{Z}) and then PU(H)BU(1), PU(H) \simeq B U(1) \,, where PU(H)PU(H) is the group of projective unitary operators on an infinite-dimensional Hilbert space.

As is mentioned in the pdf above, this is directly relevant for the categorification of line bundles, i.e. of rank-1 vector bundles, which (or rather their local semi-trivializations) are otherwise also known as line bundle gerbes.

It is not really a mystery how these are related to quantum field theory in one way or another. See for instance Mickelsson’s article Gerbes and Quantum Field Theory.

But, as you notice, the phenomenon seems to be much more general than that. Another aspect of it is how string 2-bundles are related to the quantum field theory of free fermions on the circle.

A general, good, one-sentence illumination of this phenomenon escapes me, at the moment - unfortunately.

Posted by: urs on April 23, 2007 3:13 PM | Permalink | Reply to this

Re: Some Conferences

Urs wrote:

It is true that one encounters the curious phenomenon that upon categorifying things related to vector bundles one runs into lots of structure that has a quantum-mechanical smell to it, without any quantum mechnical systems having manifestly entered the picture.

[…]

A general, good, one-sentence illumination of this phenomenon escapes me, at the moment - unfortunately.

Me too! I think we just need to keep gathering examples, and eventually some nice patterns will emerge. It has the potential for being very important! We may learn something deep about physics.

By the way, in case it’s not already obvious, when I give the aforementioned talk at the Abel Symposium, I’ll emphasize that it’s about joint work with you, Alissa, Danny and Toby. A lot of the ideas are yours. I just feel the desire to give a talk about the big picture: all the evidence that elliptic cohomology is tightly connected to higher gauge theory.

The more we get written up before that talk, the happier I’ll be. Danny and I seem very close to a nice result on classifying spaces of 2-bundles. And last Friday, Danny, Alissa and I went through your additions to our paper on canonical 2-representations of 2-groups. We should talk about whether it’s tactically better to discuss the String GString_G case (which requires topology) along with the discrete 2-group case (which doesn’t), or to make it two separate papers. The advantage of chopping it in half is that the first half is almost done, while the second half may require help from experts.

Sorry, I should probably be discussing this via email… it’s not very interesting to most people… it’s just on my mind!

Posted by: John Baez on May 8, 2007 6:11 AM | Permalink | Reply to this

Re: Some Conferences

We should talk about whether it’s tactically better to discuss the String k(G)\mathrm{String}_k(G) case (which requires topology) along with the discrete 2-group case (which doesn’t),

True. With a little luck, I will know for sure whether the canonical rep of the string-2-group exists as it seems it does by mid of June.

We may or may not discuss this then in that paper (I have the impression it will be a very simple statement in the end, not requiring long analysis) but in any case it would be very good if we could mention the result, since it will show people why the canonical 2-rep is important.

when I give the aforementioned talk at the Abel Symposium

I still haven’t made up my mind if I try to go there, too. I was asked by several people if they’ll see me there, and I would certainly love to go. Might check if there is some funding available somewhere…

By the way, in case it’s not already obvious, […]

Yes, no problem. I didn’t mean to come across in a way that you would have to emphasize this. What happened is that I was busy writing that overview of what I understand about 2-vector bundles, when that evening I wanted to relax by writing a blog entry on these conferences. After googling for the Abel symposium I came across your abstract. So I thought I might just as well post the thing I was working on here as an extended comment on that conference announcement.

Posted by: urs on May 8, 2007 12:50 PM | Permalink | Reply to this
Read the post Connections on String-2-Bundles
Weblog: The n-Category Café
Excerpt: On connections on String 2-bundles.
Tracked: June 3, 2007 4:08 PM
Read the post Generalized Geometric Langlands is False
Weblog: The n-Category Café
Excerpt: C. Teleman on a counter example to the generalised geometric Langlands conjecture.
Tracked: June 18, 2007 3:48 PM
Read the post 2-Vectors in Trondheim
Weblog: The n-Category Café
Excerpt: On line 2-bundles.
Tracked: November 5, 2007 9:46 PM
Read the post Connections on Nonabelian Gerbes and their Holonomy
Weblog: The n-Category Café
Excerpt: An article on transport 2-functors.
Tracked: August 15, 2008 8:18 PM

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