The n-Category Café

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 $\mathrm{String}G$ associated to any compact simple Lie group $G$. We describe how this 2-group is built using a central extension of the loop group $\Omega G$, and how the classifying space for $\mathrm{String}G$-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 $\rho$ of any strict 2-group $G_2$ on $\mathrm{Bim} \subset 2\mathrm{Vect}$. This allows to consider 2-vector transport 2-functors $$\mathrm{tra} : P_2(X) \to 2\mathrm{Vect}$$ which is locally $\rho$-trivializable. This gives a notion of $\rho$-associated 2-vector bundles with connection.

For the simple case $G_2 = \Sigma U(1)$ 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 $G_2 = \mathrm{String}_k(G)$. 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 $\mathrm{Bim}_{\mathrm{vN}}$.

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 $\Sigma U(1)$ on $\mathrm{Bim}$. 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.