## April 26, 2006

### Talk in Bonn on String 2-Group

#### Posted by Urs Schreiber

Next Tuesday, May 2, 2006 there is a talk

at the Bonn Math Institute ( 17:00-18:00, seminar room E (room 3, Meckenheimer Allee 160)).

Abstract.
We try to convey the main idea for
$•$ what the String-group ${\mathrm{String}}_{G}$ is
$•$ and how it is the nerve of a 2-group ${\mathrm{Str}}_{G}$
as well as
$•$ what a ${\mathrm{Str}}_{G}$-2-bundle is
$•$ and how it is ‘the same’ as a ${\mathrm{String}}_{G}$-bundle.

The first point is due to [BCSS,Henriques], which will be reviewed in section 2. The second point has been discussed in [Jurčo] using the language of bundle gerbes. In section 3 we review this, using a 2-functorial language which is natural with respect to the 2-group nature of ${\mathrm{Str}}_{G}$.

Motivation

The main motivation for the following discussion has its origin in theoretical physics.

Elementary particles with spin are described by sections of spin bundles. From the physical point of view, the necessity of a spin structure on spacetime may be deduced from a certain global anomaly for the path integral of a single, pointlike, fermion. The path integral (albeit a somewhat heuristic device) can be regarded as a single valued function on the space of configurations of the particle only if the (first and) second Stiefel-Whitney class of spacetime vanishes. In other words, if spacetime admits a spin struture.

It is possible to generalize this argument to the case where the fermion is line-like. (In theoretical physics such a hypothetical object is called a superstring.) It was found that in this case there is another obstruction, which this time is measured by the first Pontryagin class of spacetime [Killingback]. This is interpreted as saying that the loop space over spacetime admits a spin structure. In fact, this condition implies the famous (to high energy physicists) Green-Schwarz anomaly cancellation, which has been one of the main reasons why physicists considered superstrings a promising idea to pursue.

As for the pointlike fermion, this situation may be reformulated in terms of lifts of bundles. There is a topological group called $\mathrm{String}\left(n\right)$, which is a 3-connected cover of $\mathrm{Spin}\left(n\right)$. The Pontryagin class is the obstruction controlling the lift of $\mathrm{Spin}\left(n\right)$-bundles to $\mathrm{String}\left(n\right)$-bundles [StolzTeichner,Stevenson].

$\phantom{\rule{thinmathspace}{0ex}}$

For several reasons one may suspect that this situation is naturally described in terms of categorical algebra. Indeed, it can be shown that $\mathrm{String}\left(n\right)$ is nothing but the geometric realization of the nerve of a certain category with group structure, called $\mathrm{Str}\left(n\right)$ - a (Fréchet Lie) 2-group. This is the content of section 2.

Moreover, the obstruction to lifting a $\mathrm{Spin}\left(n\right)$-bundle to a $\mathrm{String}\left(n\right)$-bundle is the same as that for lifting it to a 2-bundle (gerbe) with structure 2-group $\mathrm{Str}ofn$. This is the content of section 3 [Jurčo].

Posted at April 26, 2006 8:02 PM UTC

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### Re: Talk in Bonn on String 2-Group

Hey,

Regarding how String(n)-bundles are “the same” as a Str_G-2-bundle. Maybe someone can explain a simpler version to me: Is it the case that circle-bundles are “the same” as 2-bundles for the 2-group with no 1-arrows and Z 2-arrows? What does it mean?

D.

Posted by: D. on April 29, 2006 11:36 PM | Permalink | Reply to this

### Re: Talk in Bonn on String 2-Group

Is it the case that circle-bundles are “the same” as 2-bundles for the 2-group with no 1-arrows and “\mathbb{Z}” 2-arrows?

That would seem to be one implication. I am however at the moment a little unsure about a detail of one proof. More later.

Posted by: urs on April 30, 2006 4:35 PM | Permalink | Reply to this

### Re: Talk in Bonn on String 2-Group

Everything with discrete (2-)groups should work even better than for general topological 2-groups.

As for this:

$\left(ℤ\to 1\right)$-2-bundles are “the same” as $U\left(1\right)$-1-bundles.

let’s makes this precise in the following lowbrow way:

“equivalence classes of $\left(ℤ\to 1\right)$-2-bundles are in natural 1-1 correspondence with isomorphism classes of $U\left(1\right)$-bundles.”

We may in general have some puzzlement over the correct notion of equivalence for 2-bundles, but let us hope that for a sufficiently nice space $M$ (e.g. paracompact):

1) any $\left(ℤ\to 1\right)$-2-bundle over $M$ can be described using some open cover via transition functions that give a Čech 3-cocycle valued in $ℤ$. In other words, a bunch of $ℤ$-valued continuous functions ${h}_{\mathrm{ijk}}$ on triple intersections, satisfying the obvious cocycle condition

2) two $\left(ℤ\to 1\right)$-2-bundles are equivalent iff their defining Čech cocycles are cohomologous.

Then equivalence classes of $\left(ℤ\to 1\right)$-2-bundles correspond in a natural way to elements of the Čech cohomology ${H}^{3}\left(M,ℤ\right)$.

But this is naturally isomorphic to the Cech cohomology ${H}^{2}\left(M,U\left(1\right)\right)$. (Here coefficients are in the sheaf of continuous $U\left(1\right)$-valued functions.)

And this is naturally isomorphic to the set of isomorphism classes of $U\left(1\right)$ bundles over $M$.

I think with Toby’s definition of 2-bundle 1) and 2) are true, at least for a nice space $M$.

So, I think the answer to the above question is YES.

The highbrow way of making the question precise would involve an equivalence of 2-categories, not just a 1-1 correspondence between sets of equivalence classes. I bet this works too.

Posted by: John Baez on May 1, 2006 2:33 PM | Permalink | Reply to this

### Re: Talk in Bonn on String 2-Group

Well, but it’s H^2(Z), and H^1(U(1)), that classify circle bundles… was my guess wrong?

D.

Posted by: D. on May 1, 2006 4:18 PM | Permalink | Reply to this

### Re: Talk in Bonn on String 2-Group

Above, ${H}^{2}\left(M,U\left(1\right)\right)$ denotes Čech cohomology.

Posted by: urs on May 1, 2006 4:30 PM | Permalink | Reply to this

### Re: Talk in Bonn on String 2-Group

That’s still the group that classifies Abelian gerbes.

Posted by: Aaron Bergman on May 1, 2006 4:40 PM | Permalink | Reply to this

### Re: Talk in Bonn on String 2-Group

Reduce the degree everywhere by one to get the ordinary conventions, then.

Posted by: urs on May 1, 2006 4:47 PM | Permalink | Reply to this

### Re: Talk in Bonn on String 2-Group

I don’t think this is a “conventions” problem.

If G is an ordinary group, you can regard it as a 2-group with no interesting 2-arrows. What’s a 2-bundle for, say, G = Z? I have a feeling that *this* is where H^3(Z) comes in.

D.

Posted by: D. on May 1, 2006 5:54 PM | Permalink | Reply to this

### Re: Talk in Bonn on String 2-Group

What’s a 2-bundle for, say, G = Z?

If you mean, as in your first message, a 2-bundle for the structure 2-group corresponding to the crossed module $\left(ℤ\to 1\right)$, then it is, at the cocycle level, a collection of functions

(1)${f}_{\mathrm{ijk}}:{U}_{\mathrm{ijk}}\to ℤ$

on triple intersection ${U}_{\mathrm{ijk}}$, such that on quadruple intersections ${U}_{\mathrm{ijkl}}$ we have

(2)${f}_{\mathrm{ikl}}{f}_{\mathrm{ijk}}={f}_{\mathrm{ijl}}{f}_{\mathrm{jkl}}\phantom{\rule{thinmathspace}{0ex}}.$

Hence $f$ is an element in second (yes) Čech cohomology of base space, with values in $ℤ$.

Posted by: urs on May 1, 2006 6:02 PM | Permalink | Reply to this

### Re: Talk in Bonn on String 2-Group

I just got the conventions mixed up, as I always do: a Cech n-cocycle gives data on (n+1)-fold intersections of open sets. This convention is actually good, but it always throws me.

If you reduce all the superscripts in my post by 1, what I said should make sense.

Posted by: John Baez on May 5, 2006 11:03 PM | Permalink | Reply to this

### From K-Transition Bundles to |K|-Transition Functions and Back

The proof given in 5.6 of math.DG/0510078 uses the fact that, given a crossed module bundle gerbe bundle $E\to {Y}^{\left[2\right]}\to X$ classified by ${Y}^{\left[2\right]}\to \mid K\mid$, we can always choose a gauge such that pulling back to open patches of $X$ by local sections ${U}_{\mathrm{ij}}\stackrel{{s}_{\mathrm{ij}}}{\to }{Y}^{\left[2\right]}$ we get 1-cocylces ${g}_{\mathrm{ij}}{g}_{\mathrm{jk}}={g}_{\mathrm{ik}}$ of a $\mid K\mid$ bundle on the nose, i.e. without any 2-coboundary appearing.

First I did not see how this is true. Now, I think, i do (benefiting from discussion with various people, which I’ll be glad to name – unless they do not want to be associated with what I am saying here :-).

I’ll first indicate the diagrammtic reasoning, then I give the more pedestrian version.

In the following, “1-cocycle” refers to a Čech cocycle of an ordinary bundle (i.e. ${g}_{\mathrm{ij}}$ satisfying ${g}_{\mathrm{ij}}{g}_{\mathrm{jk}}={g}_{\mathrm{ik}}$), while “2-cocycle” refers to the analogous thing for gerbes.

1) Let us choose $Y\to M$ to be a good covering $Y=\left\{{U}_{i}{\right\}}_{i}$ by open sets. One checks that the 2-morphism $t$ on p. 16 of the script given above are stritly invertible in the present situation. According to the general construction (def. 4 in these notes) this means that we may always choose the 2-morphism $\varphi$ on the same page to be in fact the identity.

But this is equivalent to saying that the Čech cocycles representing the transition bundles satisfy (under their product, wich is composition of arrows on p. 16) the 1-cocycle on the nose. Taking realized nerves everywhere should hence give us a choice of representing function ${U}_{\mathrm{ij}}\stackrel{{g}_{\mathrm{ij}}}{\to }\mid K\mid$ of the transition bundle which, too, satisfies the 1-cocycle.

Conversely, by reverse-engineering this we find that any cocycle ${U}_{\mathrm{ij}}\stackrel{{g}_{\mathrm{ij}}}{\to }\mid K\mid$ gives rise to cocycle data for $K$-crossed module transition bundles that satisfy the diagram on p. 16 with $\varphi =Id$.

Moreover, one quickly sees that a different choice of trivializations $t$ for the 2-bundle corresponds precisely to a 1-coboundary for the $\mid K\mid$-bundle cocycle ${g}_{\mathrm{ik}}$. (That’s easy in diagrams, but harder in words.)

The important point here is the strict invertibility of $t$. Had we chosen not to work at the level of Čech data, but at the level of bitorsor bundles (as indicated here), we’d deal with diagrams of precisely the same form, but now nontrivial 2-isomorphisms coming from an adjunction appear in the triangles in def 4, making it non-obvious how to obtain a strict 1-cocycle on the transition bundles.

2) Spelled out, this means the following.

We start with a 2-cocycle representing our $K$-2-bundle. We choose another good cover $Y$. Over each open patch of this cover the above 2-cocycle is cobordant to the trivial 2-cocycle. This coboundary is the $t$ from above.

Hence on double intersections we get a coboundary ${\overline{t}}_{i}\circ {t}_{j}$ from the trivial 2-cocycle to itself. One checks that such a 2-coboundary of the trivial 2-cocycle is the same as a 1-cocycle, hence represents a bundle (the transition bundle).

Moreover, the 2-coboundary $t$ has a strict inverse (with respect to the product induced by the product of $K$!). It hence follows that we get 1-cocycle for transition bundles which, under the above product, satisfy themselves a 1-cocycle, strictly.

By passing to classfying functions this 1-cocycle of $K$ crossed module bundles becomes a 1-cocycle of $\mid K\mid$-valued functions.

Posted by: urs on May 1, 2006 3:10 PM | Permalink | Reply to this

### Re: From K-Transition Bundles to |K|-Transition Functions and Back

A pdf with more details on what I outline above is now available here. A discussion of gauge transformations on both sides of the correspondence will be added in a moment…

Posted by: urs on May 1, 2006 7:13 PM | Permalink | Reply to this
Read the post How many Circles are there in the World?
Weblog: The String Coffee Table
Excerpt: On 2-torsors for suspended U(1).
Tracked: July 8, 2006 12:57 PM
Read the post Wirth and Stasheff on Homotopy Transition Cocycles
Weblog: The n-Category Café
Excerpt: Stasheff recalls an old result by Wirth on passing between fibrations and their homotopy transition cocycles.
Tracked: September 11, 2006 7:56 PM
Read the post Puzzle Pieces falling into Place
Weblog: The n-Category Café
Excerpt: On the 3-group which should be underlying Chern-Simons theory.
Tracked: September 28, 2006 3:16 PM
Read the post WZW as Transition 1-Gerbe of Chern-Simons 2-Gerbe
Weblog: The n-Category Café
Excerpt: How the WZW 1-gerbe arises as the transition 1-gerbe of the Chern-Simons 2-gerbe.
Tracked: October 29, 2006 4:47 PM
Read the post Connections on String-2-Bundles
Weblog: The n-Category Café
Excerpt: On connections on String 2-bundles.
Tracked: June 3, 2007 4:06 PM

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