### Obstructions for *n*-Bundle Lifts

#### Posted by Urs Schreiber

I am in the process of preparing some slides which are supposed to eventually provide an overview of the current state of the art of the second edge, with an emphasis on understanding String and Chern-Simons Lie $n$-algebras, the Lie $n$-algebra cohomology and the corresponding theory of bundles with Lie $n$-algebra connection in terms of the differential Lie analog of parallel transport $n$-functors, like String 2-bundles and Chern-Simons 3-bundles.

Here a Chern-Simons 3-bundle associated to a $G$-bundle should be the obstruction to lifting the $G$-bundle to a $\mathrm{String}(G)$ 2-bundle.

I know how to say this in a pedestrian way. But now I want to say it in the most elegant possible way. I want to understand how it *really*™ works.

I am too tired to do the topic justice. But all the more grateful I’d be for comments on the following considerations.

*2-bundles measuring the lift of a 1-bundle through a central extension*

Before formulating the general setup, consider the simple example of a lifting line 2-bundle.

Let $U(1) \to \hat H \to H$ be an exact sequence of groups, hence a central extension $\hat H$ of the group $H$.

Given an $H$-bundle, we may try to lift it to a $\hat H$-bundle.

The way to do this is to first of all notice that an $H$ 1-bundle is the same as a $(1 \to H)$-2-bundle.

Here I write $(G_1 \to G_0)$ for the strict 2-group coming from a crossed module of two groups $G_1$ and $G_0$.

And then to notice that the crossed module $(U(1) \to \hat H)$ is equivalent to $(1 \to H)$: $(U(1) \to \hat H) \simeq (1 \to H) \,.$ The $U(1)$ worth of automorphisms of every object in $(U(1) \to \hat H)$ exactly cancel the central part of the group of objects $\hat H$.

Therefore, every ordinary $H$-bundle is automatically a $(U(1) \to \hat H)$ 2-bundle: $H \stackrel{=}{\to} (1 \to H) \stackrel{\simeq}{\to} (U(1) \to \hat H) \,.$

More concretely, when we hit the 1-cocylce

$\array{ && \bullet \\ & {}^{\pi^*_{12} g}\nearrow && \searrow^{\pi^*_{23}g} \\ \bullet &&\stackrel{\pi^*_{13} g}{\to}&& \bullet }$

of the $H$-bundle – which is a diagram in $\Sigma H$, hence a diagram in $\Sigma (1 \to H)$ – with any of the equivalences

$(1 \to H) \stackrel{\simeq}{\to} (U(1) \to \hat H)$

(and for the cognoscenti: I am simply glossing over all smoothness or structure-ness issues for the moment)

we obtain the 2-cocycle (this “2” is my counting: some people count differently here – doesn’t matter)

$\array{ && \bullet \\ & {}^{\pi^*_{12} \hat g}\nearrow &\Downarrow^{\pi^* h}& \searrow^{\pi^*_{23}\hat g} \\ \bullet &&\stackrel{\pi^*_{13} \hat g}{\to}&& \bullet }$

(imagine this making the obvious tetrahedron 2-commute, which I won’t draw right now).

So up to this point we have simply puffed up the available 0-categorical (or 1-categorical, depending on how you count) information of an ordinary $H$-bundle to an equivalent 1-categorical (or, 2-categorical, respectively) piece of data.

The point being that the addition wiggle room we obtain this way allows some nifty moves.

Namely, what we are really interested in is whether the original $H$-cocycle can be lifted to a $\hat H$-cocycle. Now this is equivalent to asking if our $(U(1) \to \hat H)$-cocycle can be *reduced* to a $(1 \to \hat H)$ cocycle: does it factor through

$(1 \to \hat H) \hookrightarrow (U(1) \to \hat H)$ ?

That’s what we would like to know. To check if it does, we should find a measure of how this might *fail*.

In the simple case we are dealing with here, it is easy to construct this measure “by hand”: clearly, we should simply check if the image of our 2-cocycle under the projection

$(U(1) \to \hat H) \to (U(1) \to 1)$

is a trivializable $(U(1) \to 1)$ 2-cocycle or not. When this is trivializable, we may gauge it away and thereby indeed reduce our $(U(1) \to \hat H)$ 2-coycle to the desired $(1 \to \hat H)$ 2-cocycle. If not, the class of this $(U(1) \to 1)$ 2-cocycle is a good measure for the failure of the original $H$-bundle to lift to a $\hat H$-bundle.

And this class is of course nothing but the class of any line 2-bundle (which you may perhaps better know in its incarnation as a line bundle gerbe) having that 2-cocycle.

So, this way a line 2-bundle is the obstruction to lifting a principal bundle through a central extension of its structure group.

Fine. But how does this generalize? What is the power tool that achieves in general what we could achieve using a hand drill and simply guessing that we should look at the projection $(U(1) \to \hat H) \to (U(1) \to 1)$ ?

Clearly, since in general $(G_1 \to 1)$ doesn’t even exist (unless $G_1$ is abelian), the naive guess here is wrong.

*general setup*

I guess it’s clear that what is really going on is that we form a quotient:

$\array{ (1 \to \hat H) \\ \downarrow \\ (U(1) \to \hat H) &\simeq & (1 \to H) \\ \downarrow \\ (U(1) \to 1) &=& (U(1) \to \hat H)/(1 \to \hat H) }$

in a suitable sense.

And this “suitable” is where, I think, it gets tricky. Once we want to formulate this very generally.

Eventually, I would like to have a statement applicable to general $n$-groups. Like this:

For $K \to G \to B$ an exact sequence of $n$-groups, there is an $(n+1)$-group $(K \to G)$ which is equivalent to $B$ $(K \to G) \simeq (1 \to B) \,.$

Then we can form the quotient $(n+1)$-group (a cokernel of sorts) $\array{ (1 \to G) \\ \downarrow \\ (K \to G) &\simeq & (1 \to B) \\ \downarrow \\ \hat B_K &:=& (K \to G)/(1 \to G) }$

such that the obstruction to lifting a $B$-$n$-bundle through $G \to B$ is a $\hat B_k$-$(n+1)$-bundle.

(Here I haven’t said which construction the notation $(K \to G)$ means in detail. At the moment this is just notation for some $(n+1)$-group. But I expect that this needs to be read as *the mapping cone of $K \to G$*. (Compare definition 4 for the mapping cone of a morphism of two strict 2-groups.))

*special case: inner automorphism $(n+1)$-groups*

As a fun special case, whose understanding probably helps understanding the general situation, consider sequences of $n$-groups whose left morphism is the identity:

$G_{(n)} \stackrel{\mathrm{Id}}{\to} G_{(n)} \to 1 \,,$

where 1 is the trivial $n$-group.

Here clearly $(G_{(n)} \stackrel{\mathrm{Id}}{\to} G_{(n)})$ is precisely the mapping cone, hence, according to the considerations in the inner automorphism 3-group of a strict 2-group, we take $(G_{(n)} \to G_{(n)}) := \mathrm{INN}_0(G_{(n)}) \,.$

Indeed, this is equivalent to the trivial $n$-group $\mathrm{INN}_0(G_{(n)}) \simeq 1$

by general properties of tangent categories.

If $G_{(n)}$ is “sufficiently abelian”, we do know what the cokernel of $\array{ G_{(n)} \\ \downarrow \\ \mathrm{INN}_0(G_{(n)}) }$ should be: $\Sigma G_{(n)}$.

(If $G_{(n)}$ is not “sufficiently abelian”, I suspect we find the quotient has to be an $n$-group for potentially very high $n$: related to the highest degree of $\mathrm{Lie}(G_{(n)})$-invariant polynomials.To see what I am thinking of here, you might go to the part titled *The algebra of $g_{(n)}$-invariant polynomials* in those slides. This is a subtle aspect which I would dearly appreciate comments on.)

Given that, as I kept emphasizing, the $(n+1)$-curvature of a $G_{(n)}$ $n$-transport takes values in $\mathrm{INN}(G_{(n)})$, we seem to be able to interpret this, in the present context, in terms of the following somewhat entertaining slogan:

The $(n+1)$-curvature of an $n$-transport is the lifting $(n+1)$-bundle with connection of the lift of an $n$-bundle with

trivialstructure group to $G_{(n)}$.

I need to think more about how to make this precise. But this looks like it could actually be a very useful way to look at the construction illustrated by that “diagram movie with subtitles” in the part “$n$-categorical background” of those slides.

*finally: Lie $n$-algebra*

I am thinking that the *really*™ good way to handle Chern-Simons 3-bundles along the lines of the second edge as described in those slides is to pretty much follow the above construction, with Lie $n$-algebras instead of Lie $n$-groups. Then by expressing these Lie $n$-algebras dually in terms of their dual quasi-free differential graded commutative algebras (qDGCAs), the $(K \stackrel{t}{\to} G)$-construction should be precisely the mapping cone construction of the corresponding $t^*$ on the level of qDGCAs.

I had energy to make some quick consistency checks that this indeed gives the right answer. For instance when applied to the exact sequence

$\mathrm{Lie}(\Sigma U(1)) \to g_{\mu_k} \to g$

for $g$ an ordinary Lie algebra and $g_{\mu_k}$ the corresponding Baez-Crans type String Lie 2-algebra, it seems that the mapping cone construction indeed correctly eliminates the vanishing of the first Pontryagin class in the collection of characteristic classes of the corresponding $g_{(n)}$-bundles. That’s exactly what should happen for the Chern-Simons 3-bundle which measure the obstruction of the lift from a $g$-bundle to a $g_\mu$-bundle to exist, which is nothing but this Pontryagin class.

But I need to spell this out in more detail after I have recharged my batteries.

Meanwhile, all comments are very welcome.

## Re: Obstructions for n-bundle lifts

I believe I made some progress with understanding this issue, at the level of Lie $n$-algebras.

In the file

I describe:

- how the (strict) cokernel of a morphism of semistrict Lie $n$-algebras $f_{(n)} \to g_{(n)}$ looks like; how it is in general a Lie $n'$-algebra for $n' \gt n$

- how the cokernel of the canonical embedding $g_{(n)} \to \mathrm{inn}(g_{(n)})$ is, as a special case of a general statement, indeed the Lie $n'$-algebra which I have started calling $b g_{(n)}$, characterized by the fact that its dual is generated from all the classes of invariant $g_{(n)}$ polynomials

- how in special cases the cokernel of morphisms $f_n \to g_n$ of Lie $n$-algebras is itself a Lie $n$-algebra and then equivalent to the mapping cone Lie $(n+1)$-algebra $(f_n \to g_n)$

- how all this can be used to determine the codomain of obstructions to extensions of structure Lie $n$-algebras of $g_{(n)}$-bundles .

I list a couple of examples illustrating this, containing also the Lie-version of the lifting gerbe construction described in the entry above.

The discussion of the Chern-Simons 3-bundles as obstruction to lifts through the string extension $\mathrm{Lie}(\Sigma U(1)) \to g_\mu \to g$ is then a straightforward generalization of this discsussion, left as an exercise (to both, reader and author ;-).