## August 17, 2005

### CFT, Gerbes and K-Theory in Oberwolfach, V

#### Posted by urs

Wednesday in Oberwolfach. The obstructions for the sun to shine finally lifted and we had something that even people from Australia would recognize as a summer day.

The workshop now has a webpage on which links to slides and other background information will eventually appear:

Workshop Gerbes, twisted K-theory and conformal field theory

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 $E\to {Y}^{\left[2\right]}$ 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 $E\stackrel{U\left(1\right)}{\to }{Y}^{\left[2\right]}$ being a $U\left(1\right)$-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 $\mathrm{Aut}\left(H\right)$. In a slight generalization of this construction this is replaced, more generally, by any group $G$ with a homomorphism $G\to \mathrm{Aut}\left(H\right)$ such that $\left(G,H\right)$ 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 $U\left(N\right)$-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

(1)$〈\mathrm{exp}\left({\int }_{\Sigma }B\right)〉$

of ‘vertex operators’

(2)$B\in {\Omega }^{2}\left(M,{V}^{\otimes 3}\right)$

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

(3)$\left[\mathrm{Hom}\left(M,\mid {G}_{2}\mid \right)\right]\phantom{\rule{thinmathspace}{0ex}},$

where $M$ is base space and $\mid .\mid$ 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 $m>n$) 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 ${Q}_{A}=Q+\mathrm{ik}\int d\sigma \psi \left(\sigma \right)\cdot A\left(\sigma \right)$, where $A\in {\Omega }^{1}\left({S}^{1},\mathrm{Lie}\left(G\right)\right)$ 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.

Posted at August 17, 2005 8:18 PM UTC

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### Local gerbes?

Can one give a local description of a gerbe?

We can define manifolds and bundles by covering the manifold with local charts, and require that the transfer functions satisfy a 1-cocycle condition. I understand that gerbes generalize this construction to something satisfying a 2-cocycle condition instead. However, in a single chart we can give a much more concrete description: a manifold is R^n with some coordinates and a section of a bundle is a function of these coordinates. Essentially, we can phrase everything in the language of tensor calculus.

Is there an analogous 2-tensor calculus that describes gerbes locally? It should be, unless gerbes are intrinsically non-local objects, but I have never seen anything like that.

### Re: Local gerbes?

However, in a single chart we can give a much more concrete description: a manifold is ${R}^{n}$ with some coordinates and a section of a bundle is a function of these coordinates.

If I understand correctly you are saying that locally a fibre bundle with typical fibre $F$ just looks like ${U}_{i}×F$ and that sections are locally just functions in $\mathrm{Mor}\left({U}_{i},F\right)=\mathrm{Map}\left({U}_{i},F\right)$. You are asking for the gerbe-analogue of that, right?

In as far as a 2-bundle can be understood as a gerbe, the analogue statement is that the gerbe locally looks like ${U}_{i}×F$, where $F$ is now a category (a 2-group, for instance). Sections now locally are again in $\mathrm{Mor}\left({U}_{i},F\right)$, which now however means that they are functors ${U}_{i}\to F$.

As long as ${U}_{i}$ is an ordinary manifold (i.e. a smooth category with all morphisms identities) such functors are of course again nothing but ‘functions’ again. But the point is that they do have natural transformations between them. So we get a (functor-)category $C\left({U}_{i}\right)$ of sections for each open set ${U}_{i}$.

Gluing these together on double overlaps ${U}_{i}\cap {U}_{j}$ has therefore a little more freedom that just gluing functions, and this is what gives rise to the enriched cocycle conditions in the cocycle description of the gerbe.

In fact, the stack-theoretic definition of a gerbe takes this section-point-of view as its starting point. A gerbe, in that definition, is the categorification of a sheaf (called a ‘fibred category’) with some additional properties, namely an assignment of categories $C\left({U}_{i}\right)$ (groupoids, actually) to open sets ${U}_{i}$ and restriction functors

(1)$C\left({U}_{j}\right)\stackrel{{\mid }_{\mathrm{ji}}}{\to }C{U}_{i}$

whenever

(2)${U}_{i}\subset {U}_{j}$

such that iterated restriction ${U}_{k}\to {U}_{j}\to {U}_{i}$ is isomorphic to direct restriction ${U}_{k}\to {U}_{i}$.

In the bundle gerbe way of looking at gerbes this sort of picture of sections and the like is pretty much obscured (as far as I am concerned, at least).

Posted by: Urs on August 18, 2005 2:44 PM | Permalink | Reply to this

### Re: Local gerbes?

There are various proofs that bundle gerbes are gerbes, but the “best” version is (IMO) in

Bundle gerbes: stable isomorphism and local theory (Murray and Stevenson) math.DG/9908135

this comes at it from the bundle gerbe side.

In outline, the authors construct a fibred category consisting of all local trivialisations of the bundle gerbe over open sets in the base $M$. Trivialisations for bundle gerbes are $U\left(1\right)$ bundles that are lifted to bundle gerbes (thinking in just the abelian case here). It is a well known fact that the collection of $U\left(1\right)$-bundles on a manifold forms the trivial (neutral) $U\left(1\right)$-gerbe. This fibred category is then shown to be a gerbe.

This description fits in nicely with the point made by Breen in these notes on 1- and 2-gerbes, that we can look at the “frame-stack” of a gerbe, much like we can consider the frame bundle of a vector bundle, instead of the bundle itself. Really this is thinking of the gerbe as a twisted version of a neutral gerbe, much as a manifold is a twisted version of ${R}^{n}$. Really when we consider a manifold, it is the existence of a non-trivial structure sheaf (= collection of smooth functions to Euclidean space) that makes the manifold globally different to ${R}^{n}$.

D

Posted by: David Roberts on August 19, 2005 4:57 AM | Permalink | Reply to this

### Re: Local gerbes?

There are various proofs that bundle gerbes are gerbes

Maybe the most direct relation between a bundle gerbe $B$ and the coresponding gerbe ${G}_{B}$ is that $B$ (being a groupoid) is a presenting groupoid of ${G}_{B}$ in the sense as we have discussed recently.

At least that’s what Danny seems to be telling me. I can’t claim to be able to demonstrate this, currently.

Posted by: Urs on August 19, 2005 7:04 PM | Permalink | Reply to this

### Re: Local gerbes?

I guess that you already answered my question, but let me see if I understood it right. A section of a 1-bundle is locally a function

f(x) = f_a(x) e^a,

where the e^a form a basis for the fiber and x is a point in the base space. So a similar formula should apply to 2-bundles, with the e^a a basis for a category? Perhaps one can view sections as functions of x modulo some equivalence relation, which has to satisfy a coherence law on triple overlaps?

### Re: Local gerbes?

where the ${e}^{a}$ form a basis for the fiber

You are thinking of the special case of a vector bundle, where the typical fibre $F={K}^{n}$ (say) is a vector space over some field $K$. More generally, the fibre can be about anything you like. In particular it can be a group $G$, in which case you cannot write local sections the way you did here.

a basis for a category?

Certain categories might actually have a ‘basis’ in this sense, namely those that are like categorified vector spaces over somthing.

For instance the 2-vector spaces defined in HDA VI are categories internalized in $\mathrm{Vect}$. This means that these are like ordinary categories with a set of objects and a set of morphisms, only that these sets really are vector spaces and source, target and composition maps between these vector spaces are linear maps.

So if you want to consider a 2-bundle whose typical fibre looks like a category which is a 2-vector space in this sense, then, yes, I assume you can choose a basis for the space of morphisms of that category and express a 2-section of such a 2-bundle locally in terms of such a basis the way you indicated.

Note though, as I have remarked before, that the concept of a categorified vector bundle (and, more generally, that of a categorofied associated bundle) is unfortunately still much less studied than that of a categorified principal bundle (whose typical fibre is (isomorphic to) a group).

Posted by: Urs on August 20, 2005 1:40 PM | Permalink | Reply to this

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