The String Coffee Table

Today was all about gerbes and black forest cake.

In the first morning session Paolo Aschieri reviewed his work with Brano Jurčo and Luigi Cantini

Paolo Aschieri, Luigi Cantini & Branislav Jurco
Nonabelian Bundle Gerbes, their Differential Geometry and Gauge Theory
hep-th/0312154

on the nonabelian version of Murray’s bundle gerbes, including some of the string theory motivation that is discussed in

Paolo Aschieri & Branislav Jurco
Gerbes, M5-Brane Anomalies and $E_8$ Gauge Theory
hep-th/0409200.

A bundle gerbe is a differential geometric realization of a gerbe. It consists of a bundle $Y \to M$ over base space (which you should think of as the bundle of paths from the fiber base to some fixed point) together with a bundle $E \to Y^{[2]}$ over the fiber product of $Y$ with itself (which should be thought of as the bundle of pair of paths between a fixed point and the fiber base) such that there is a groupoid structure on $E$ by fiberwise composition.

An abelian bundle gerbe has $E \overset{U(1)} \to Y^{[2]}$ being a $U(1)$-bundle and is by now a very popular structure useful in many ways. What Aschieri, Cantini & Jurčo did was to figure out how the things one does with bundle gerbes generalize once $E$ is taken to be a nonabelian principal bundle.

First of all, in order to preserve the groupoid structure on $E$ one has to be able to act with some group $H$ from the left as well as from the right on $E$. Such a gadget is called an $H$-bibundle. Since $H$ acts transitively and freely on each fiber the left and right action must be related by a group automorphism in $\mathrm{Aut}(H)$. In a slight generalization of this construction this is replaced, more generally, by any group $G$ with a homomorphism $G \to \mathrm{Aut}(H)$ such that $(G,H)$ form a (‘Whitehead”-)crossed module of groups. This is nothing but the 2-group which appears in the context of 2-bundles.

This is the easy part. Things become more subtle when the notion of connection and curving are generalized from abelian to nonabelian bundle gerbes. The stronger structure of a bibundle requires that one uses some not-quite-obvious modifications of the usual definitions of a gerbe connection 1- and curving 2-form.

When done correctly, one finds that the resulting structure has the same cocycle description as that found originally by Breen & Messing.

Possibly interestingly for string physics, one can also study twisted nonabelian bundle gerbes, which (like twisted bundles) are really abelian bundle gerbes of still one degree higher, i.e. abelian 2-bundle gerbes. There is a 2-gerbe analog of the well-known argument that the coupling of the 1+1D string to the KR 2-form 1-gerbe connection leads to its endpoints being coupled to a nonabelian (twisted) 0-gerbe (=bundle on the D-brane). The analog suggests that the coupling of the 2+1D M2 brane to the supergravity 3-form 2-gerbe connection (if it really is one) implies that the M2’s boundary string couples to a nonabelian 2-form 1-gerbe connection on M5 branes.

I have written about this situation several times before, but am not sure if any real progress has been achieved in the meantime. I had some discussion of this point with Jarah, who knows probably all there is known of the string aspects relevant to this question, but the situation remains murky. Most people seem to agree that this is the most plausible thing, but a direct ‘proof’ (string-theory-wise) has not materialized yet.

In this context there was also (again) some discussion concerning the point how nonabelian gerbes could account for the $N^3$-scaling behaviour on 5-branes. I believe it is important to realize that (as far as I can see, at least), this is a question concerning the classes of 2-connections that we can put on a gerbe/2-bundle.

Noting that the analogous $N^2$-scaling behaviour on D-branes is a direct consequence of the fact that a local connection 1-form of a $U(N)$-bundle takes values in rank-2 tensors, it is kind of suggestive to speculate that $N^3$-scaling comes from a rank-3-tensor valued 2-form. The question would then be if there is any 2-connection taking values in a Lie-2-algebra that would (locally) somehow give rise to something like this.

In a series of recent papers E. Akhmedov noted that such a situation naturally occurs when we consider correlation functions

of ‘vertex operators’

in a Fukuma-Hosono-Kawai 2T TFT. If one could identify the underlying 2-group $G_2$ such that $G_2$-surface holonomy in a nonabelian $G_2$ gerbe reproduces these correlators, such gerbes with such 2-connection would be an interesting candidate for the description of M5-branes.

I have once discussed some details that the identification of such a $G_2$ would involve here.

The next talk was by Brano Jurčo, the punchline of which I had already mentioned last time. Brano considers simplicial bundles and in particular simplicial bibundles, uses the fact that these really come from 2-groups $G_2$ and then shows how their associated bundle gerbes (without connection and curving) are classified by

where $M$ is base space and $|.|$ denotes the geometric realization of the nerve of a 2-category. Recall again from Duskin that the nerve of a $n$-category $C$ is the simplicial space obtained by considering each $m$-morphism (including all identity $m$-morphisms for all $m \gt n$) as an $m$-simplex.

I am hoping to write up something with Brano generalizing this to fake flat gerbes with conenction and curving as described in the last entry.

The last talk today was by Jouko Mickelsson on ‘Twisted K-theory and the index on $G$’. As far as I am aware most of the content can be found in the nice overview

J. Mickelsson
Families index theorems in supersymmetric WZW model and twisted K-theory
hep-th/0504063.

The idea is to take the supercharge of a WZW model and form from it a family of Fredholm operators on the string’s super-Hilbert space by setting $Q_A = Q + ik \int d\sigma \psi(\sigma) \cdot A(\sigma)$, where $A \in \Omega^1(S^1, \mathrm{Lie}(G))$ is sort of a connection on the spatial part of the string. These operators transform equivariantly under conjugation by projective unitary operators on $H$ and one gets a bundle of such operators. The homotopy classes of sections of this bundle give K-theory on $G$.

I was wondering if there is some physical way to think of this construction. Somehow the statment is that a family of non-conformal CFT’s in the vicinity of the WZW string knows all about the conserved charges of boundary conditions of the string in the WZW background. Is there a way to understand this from a heuristic string-physics point of view? Can $A$ be regarded as the pull-back of a connection on a brane in the target $G$? Do the $Q_A$ define boundary states for these branes, maybe? I have no clue. Does anyone?

Ah right, and the rest of the day was spent hiking in the black forest, visiting a black forest restaurant, ordering black forest cake, realizing that the black forest restaurant had run out of black forest cake and hiking back through black forest to the institute.