The n-Category Café

Urs wrote:

Shouldn’t the real characteristic classes of a 2-bundle come from invariant expressions involving its 3-form curvature?

That’s an obvious guess — but maybe the 2-form fake curvature plays a role too!

Anyway, I was thinking that once one has defined invariant polynomials on Lie $n$-algebras this way, there would be an obvious way to generalize Chern classes etc, and in fact get a notion of characteristic classes of $n$-bundles.

Right, that’s exactly what I’m hoping.

The new twist is that now I know for other reasons what the ring of characteristic classes should be! At least I’m pretty sure, based on what I was told in Oslo — I need to check some stuff. But, it seems very plausible that the ring of (real) characteristic classes for $String_k(G)$-2-bundles is just the ordinary ring of characteristic classes for $G$-bundles, modulo the 2nd Chern class:

$$\mathrm{H}^*(\mathrm{B}|String_k(G)|,\mathbb{R}) \cong \mathrm{H}^*(\mathrm{B}G, \mathbb{R})/[c_2]$$

Note, this means that for $G = SU(n)$ we’ll get a polynomial ring on generators $c_3,\dots, c_n$ with $c_i$ of degree $2i$.

Now, this seems quite different from the guess you seem to be making, namely that all chacteristic classes are built from the 3-form curvature, hence have degrees that are multiples of 3.

So, I believe something interesting is going on here. Perhaps I’m just making some mistake, but I think it’s something more interesting than that. I need to pack now, but we can talk a bit about it in Vienna.

Coming at it from a different perspective:

arXiv:0712.2069
Title: Cohomology of Lie 2-groups
Authors: Gregory Ginot, Ping Xu

I believe I understand now what’s going on:

let $g_\mu$ be a Lie $n$-algebra of Baez-Crans type, coming from the $(n+1)$-cocycle $\mu$ on $g$.

Let $\mathrm{inn}(g_\mu)$ be the corresponding Lie $(n+1)$-algebra of inner derivations.

The Koszul dual qDGCAs I write $$(\wedge^\bullet ( s g_\mu^*), d_{g_\mu} )$$ and $$(\wedge^\bullet ( s g_\mu^* \oplus ss g_\mu^*), d_{g_{\mathrm{inn}(g_\mu)}} )$$ respectively. They sit inside the weakly short exact sequence $$(\wedge^\bullet ( s g_\mu^*), d_{g_\mu} ) \leftarrow (\wedge^\bullet ( s g_\mu^* \oplus ss g_\mu^*), d_{g_{\mathrm{inn}(g_\mu)}} ) \stackrel{p^*}{\leftarrow} \wedge^\bullet ( ss g_\mu^*)$$ which plays the role of differential forms on the universal “$G_\mu$” $n$-bundle $$G_\mu \to \mathrm{INN}_0(G_\mu) \to \Sigma G_\mu \,.$$

A characteristic class is anything in the rightmost part which is $d_{\mathrm{inn}(g_\mu)}$-closed once pulled back along $p^*$, modulo things that are $d_{\mathrm{inn}(g_\mu)}$-exact.

That’s the setup, just to remind you all.

So, let’s see what happens in the case that $g$ is simple and that $$\mu = \langle \cdot , [\cdot,\cdot]\rangle$$ is the canonical 3-cocycle built from the Killing form $\langle \cdot, \cdot \rangle$.

Then $\wedge^\bullet ( ss g_\mu^*)$ is generated from the ordinary curvature 2-forms with values in $g$, together with a new generator, to be called $$c$$ in degree 3 which is to be thought of as the 3-form curvature of our $g_\mu$-2-connection.

Everything behaves essentially as for just the ordinary Lie algebra cohomology for $g$, only that now we have on top of that the relation $$d c = 3 C_{abc} t^a \wedge t^b \wedge r^c \,,$$ where I have fallen back to using bases: $\{t^a\}$ is a basis of $s g^*$ and $\{r^a\}$ (the 2-form curvature) the corresponding one on $s s g^*$. $C_{abc}$ are the components of the 3-cocycle in that basis.

So the point is: $c$ is not $d_{\mathrm{inn}(g_\mu)}$ closed and in fact no nontrivial exterior product containing $c$ can be closed.

This would imply that the characteristic classes of $g_\mu$ are just those of $g$, coming from ordinary invariant polynomials $$k_{a_1 \cdots a_n} r^{a_1} \wedge \cdots \wedge r^{a_n} \,.$$

While it is true that this are all the $d_{\mathrm{inn}(g_\mu)}$-closed elements, the presence of $c$ now makes some of them become exact!

Namely, it is eacy to check that with $$k = k_{ab} r^a \wedge r^b$$ the Killing form invariant degree 2-polynomia, regarded as an element of $\wedge^\bullet (ss {g_\mu}^*)$ we have $$k = d_{\mathrm{inn}(g_\mu)}( c + 2 k_{ab} t^a \wedge r^b ) \,.$$

So, this means the Killing form is now exact and “drops out of the cohomology”. Hence $\langle F_A \wedge F_A\rangle$ is no longer a representative of a characteristic class.

I think it is also clear that the Killing form is the only form being killed (hahah…) this way.

I wrote:

I believe I understand now what’s going on:

Lest my above ramblings remain entirely unintelligible, I have tried to write this up more cleanly in a pdf:

Lie $n$-Algebra Cohomology

This contains mostly the old stuff – polished(!) – and now also includes a new part (last part) which

- defines cocycles, invariant polynomials and transgression elements for arbitrary Lie $n$-algebras

- discusses the ring of invariant polynomials for the String Lie 2-algebra

- seems (unless I am making a mistake) to confirm exactly the thing about the classes of a $\mathrm{String}(G)$-bundle being those of the underlying $G$-bundle modulo the second one that John Baez was talking about above.

In case you are hesitant to “leave the Café” by following the above link, here is the abstract:

Abstract: Ordinary Lie algebra cohomology of a Lie algebra $g$ has a nice reformulation in terms of the Koszul dual differential algebra of the Lie 2-algebra of inner derivations of $g$. For every transgressive degree $n$ element in $g$-cohomology there is a short exact sequence of Lie $n$-algebras. These are characterized by the fact that $n$-connections taking values in them come from the corresponding Chern-Simons forms and characteristic classes. A straightforward generalization of this construction yields a notion of cohomology, invariant polynomials and transgression elements for arbitrary Lie $n$-algebras. And in turn, each such element of degree $d$ induces a new Lie $\mathrm{max}(n,d)$-algebra. The invariant polynomials of Lie $n$-algebras $g_{(n)}$ should correspond to characteristic classes of $n$-bundles for Lie $n$-groups $G_{(n)}$ integrating these. While the general theory of these $n$-bundles is not well understood yet, we discuss that the invariant polynomials on the String Lie 2-algebra do reproduce what one expects from topological reasoning.

I am wondering if we are on the verge here of actually being able to get our hands on all these $n$-bundles by doing suitable pullbacks of the universal one.

Namely, it’s supposedly true that for $g_{(n)}$ any Lie $n$-algebra (even any Lie $n$-algebroid, but let’s not get into that for the moment), the Koszul dual qDGCA of $$\mathrm{inn}(g_{(n)})$$ plays the role of the differential algebra of differential forms on the total space of the universal principal $G_{(n)}$-bundle.

Here I write $G_{(n)}$ for the Lie $n$-group which I imagine integrates $g_{(n)}$. I know that in general that integration is a tricky issue. But that’s the point here: we may not ever need to do it.

Namely, the sequence $$G_{(n)} \to \mathrm{INN}(G_{(n)}) \to \Sigma G_{(n)}$$ which would express the universal $G_{(n)}$-$n$-bundle is supposedly represented differentially by the corresponding Koszul dual of its Lie version: $$(\wedge^\bullet (s g_{(n)}^*), d_{g_{(n)}}) \leftarrow (\wedge^\bullet (s g_{(n)}^* \oplus s s g_{(n)}^*), d_{\mathrm{inn}(g_{(n)}})) \leftarrow \wedge^\bullet (s s g_{(n)}^*) \,.$$

Since, in principle, we get every $G_{(n)}$-$n$-bundle on a space $X$ by pulling back the former sequence along a classifying morphism $$Y^{[2]} \to \Sigma G_{(n)}$$ (for $Y \to X$ a good covering of $X$ and $Y^{[2]}$ the corresponding groupoid) I am thinking that we should be able to get our hands on the differential version of this.

My first, naive, idea was to say: okay, so let’s look at morphisms of free graded commutative algebras $$\Omega^\bullet(X) \leftarrow \wedge^\bullet (s s g_{(n)}^*) \,.$$ These look like they should be the differential analog of the classifying maps $Y^{[2]} \to \Sigma G_{(n)}$.

Then one could imagine pondering pushing out the second sequence above along this map.

For the simple case that the Lie $n$-algebra is that of the abelian Lie $n$-group $\Sigma^{n-1} U(1)$ this actually almost seems to make sense: in this case such a morphism $$\Omega^\bullet(X) \leftarrow \wedge^\bullet (s s g_{(n)}^*)$$ is precisely the choice of an $(n+1)$-form on $X$, and nothing else. So, for instamce, for ordinary $U(1)$, this would mean that the “classifying map is” a 2-form on $X$. And that’s in fact almost the right answer! (It needs to be a 2-form with integral periods.)

So maybe there is something roughly right about this, but it can’t be quite as simple.

Does anyone see what else would be the right thing to do?

One reason why the above idea sort of works for the abelian case but fails more generally is that in the abelian case the identity is an invariant polynomial. So maybe we should instead think about pushing forward along sequences built from Chern Lie $n$-algebras, which encode all the characteristic classes.

Here I mean something like this:

a plain Lie algebra $g$ valued 1-form on $X$ is a qDGCA morphism $$\Omega^\bullet(X) \leftarrow \mathrm{inn}(g)^* \,.$$ To extract its $n$th class $k$, we precompose this simply with the canonical map $$\mathrm{inn}(g)^* \leftarrow \mathrm{ch}_k(g)^* \,.$$ Then the composite $$\Omega^\bullet(X) \leftarrow \mathrm{inn}(g)^* \leftarrow \mathrm{ch}_k(g)^*$$ is a Lie $n$-algebra valued connection, whose top level curvature form is precisely the given class of our original connection.

If we like, we can extract this even more explicitly by precomposing once more with the canonical $$\mathrm{ch}_k(g)^* \leftarrow \mathrm{Lie}(\Sigma^{n}U(1))^* \,.$$ The composite $$\Omega^\bullet(X) \leftarrow \mathrm{inn}(g)^* \leftarrow \mathrm{ch}_k(g)^* \leftarrow \mathrm{Lie}(\Sigma^{n}U(1))^*$$ is precisely one $n$-form, nothing else, and that $n$-form is nothing but the given class obtained from our chose connection.

So, you see, this is quite similar to having a morphism $$\Omega^\bullet(X) \leftarrow \wedge^\bullet (s s g_{(n)}^*)$$ as in the first naive attempt, only that it makes more invariant sense, somehow.

But now I am not sure how to contiunue from that point on.

There is something going on here which is probably important, but the meaning of which still escapes me:

it’s funny how $\mathrm{inn}(g_{(n)})$ appears crucially in all these constructions, but in fact in two different roles:

on the one hand it absorbs the curvatures of our connections, and allows them to be non-flat in the first place.

On the other hand, it plays the role of forms on the universal $G_{(n)}$-bundle itself.

Somehow I feel I should somehow mix these two statements, throw in a pushout or two, cook it for a while, and get out a nice cool theory of $n$-bundles with connections all in terms of just qDGCAs.

I thought about this until late last night. But I still don’t see it.

I’ve never seen it worked out fully, but it might could follow from the following:

in the smooth case, there is a construction of a universal G-bundle WITH connection, but the only proofs I’ve seen are rather brute force and via ‘finite’ approximations. Then indeed the holonomy of loops should give the desired map Ω BG → G. It might even be a morphism of smooth monoids, whereas, without connection, I would expect only an A_∞-map.

I think I can outline a natural construction of a G-connection on the universal bundle over BG.

First, consider X an arbitrary manifold equipped with a principal G-bundle P. Then G-connections on P are sections of a certain _bundle of affine spaces_ over P. If someone doesn’t know how it works, I can explain it. Anyway, denote the total space of this bundle C(P).

Now take X a “manifold” equipped with a principal G-bundle P with contractible total space (i.e. X is a realization of BG). Usually it has to be an infinite-dimensional space so it’s not really a manifold, but we need some kind of smoothness since we’re gonna do connections on this guy.

C(P) is a bundle of affine spaces, hence it’s homotopically equivalent to its base X. Hence it’s also a realization of BG. Denote pi: C(P) -> X the projection. The principal G-bundle P’ on C(P) is the pull-back of P by pi. P’ has a natural connection A. Why? To define A at a point x of C(P) we need to choose a point of C(P’)_x. However, we have a canonical “pull-back” morphism from C(P)_pi(x) to C(P’)_x and C(P)_pi(x) has a canonical point: x itself! Its pull-back is the desired point of C(P’)_x.

Now consider

X = pt / G

In this case the total space of P is the point, but we have a non-trivial stack C(P). The same construction as before yields a canonical G-connection on C(P). I believe that C(P) is thus the stack classifying principal G-bundles equipped with a connection.

Cool! As you know, this is related to stuff I’ve talked about (see page 13 here). But you’re right, it seems new.