The n-Category Café

But what is needed to move from an unstructured “zoo” or “menagerie” to a more complete classification?

Right, very good question.

Before commenting on that, let me first put this good question in the historical perspective by quoting the very first sentence of

M. Daily, T. Lada, A finite dimensional $L_\infty$-algebra example in gauge theory (ps)

(from 2005) where it says:

Although $L_\infty$ algebras (or sh Lie algebras) have been objects of much research during the past several years, concrete examples of these structures remain somewhat elusive.

Given that state of the art, the next best thing one should hope to obtain is certainly a good inventory of examples.

And more: we don’t just want arbitrary examples, we want some that actually occur in nature – and we want to gain something for our other tasks by identifying Lie $n$-algebras where they have previously remained unidentified (or addressed, irritatingly, as “soft group manifolds”).

Daily and Lada in the above paper end their introduction with the sentence

We leave it as a challenge to the physicists to develop a physical model whose gauge transformations are described by this algebraic examples.

So that’s a solution looking for a problem. We’d rather have it the other way round.

I hope this makes it look more pardonable that the title of our document contains the word “zoo” without it being preceeded by the words “taming of the” (as in the famous Moore-Seiberg paper).

That said, I would like to emphasize that the bulk of the paper goes into considerably more structural investigations than title might suggest.

While lots and lots clearly remains to be better understood, we do amplify a couple of striking patterns that do seem to have significance. Among them clearly is the $\mathrm{inn}$-construction, as discussed in the section “Main Results”.

On the other hand, as mentioned in the section “open problems” there are indications that there is a major structural issue here which we appear to be scratching the surface of – but not more.

Well I am being a little vague here, and that alone is of course part of the answer to your problem: much more needs to be understood, certainly.

But to close with something non-vague, I should at least recall that for semistrict Lie 2-algebras, John Baez and Alissa Crans proved a classification result saying that these are classified by

- a Lie algebra $L$

- an $L$-module $M$

- an element in $H^3(L,M)$.

This parallels a similar result for Lie 2-groups, given by John Baez and Aaron Lauda.

Urs wrote:

What […] objects get their interesting properties from entities in the zoo?

Quantum field theories.

And more generally: anything that has continuous symmetries, symmetries between symmetries, symmmetries between symmetries between symmetries… and so on to the $n$th level, will have a Lie $n$-algebra associated to it!

Just above proposition 5 you mention $\mathrm{INN}(G)$ and I just realised that it is the groupoid version of $E G$, namely the realisation of it is Milnor’s universal bundle.

Yes. $\mathrm{INN}(G)$ is the strict 2-group coming from the Lie crossed module $$\mathrm{id} : G \to G \,.$$ Since that is equivalent to the trivial 2-group, the realization of its nerve is a contractible space, and indeed one finds that $$\array{ |(G \stackrel{\mathrm{Id}}{\to} G)| &\simeq& E G \\ \downarrow \\ |\Sigma G| &\simeq& B G }$$ is the universal $G$-bundle.

For a general crossed module $$(H \to G)$$ one finds that the realization of the nerve, as a category, of the corresponding 2-group $$|(H \to G)|$$ is the classifying space of “$H$-bibundles relative to $G$” (you know what I mean, everybody has another name for these).

That was noticed at least by Branislav Jurčo in math.DG/0510078.

what is the corresponding construction for a 2-group?

All I know is that for any 2-grpup $G_{(2)}$ we canonically have the 3-group $\mathrm{AUT}(G_{(2)})$ which has a sub-3-group $\mathrm{INN}(G_{(3)})$.

Whether and how this, however, generalizes the relation to classifying spaces above I don’t know.

David,
You’re quick - or else the current version is more transparent. I struggled for quite a while at the wrong aspects
before recognizing inn as contractible;
in fact, generated by a mapping cone.
Then I finally recalled that for a connected group, inner automorphisms are homtopic to the identity.

As for EG, see also the section on the Weil algebra - which predates Milnor’s construction.

jim

This must be related to what Alissa and I did. We classified all Lie $n$-algebras with a Lie algebra $L$ as objects and a representation $V$ of $L$ as $(n-1)$morphisms, in terms of the Lie algebra cohomology group

$$H^{n+1}(L,V).$$

In particular, when $L = Lie(G)$ with $G$ compact connected semisimple, and $V = \mathbb{R}$ is the trivial representation, these cohomology groups are just the same as the cohomology groups of the Lie group $G$ regarded as a space. These cohomology groups are well-known — you can look them up in tables (see page 11 here). So, we get a bunch of Lie $n$-algebras.

For example, for $G = SU(k)$, we have

$$H^{n+1}(Lie(G), \mathbb{R}) = \mathbb{R}$$

when $n = 2, 4, 6, \cdots, 2k-2$ and

$$H^{n+1}(Lie(G), \mathbb{R}) = 0$$

otherwise. So we get a 1-parameter family of interesting Lie $n$-algebras with $su(k)$ as 0-chains and $\mathbb{R}$ as $(n-1)$-chains precisely for $n = 2,4,6,\cdots, 2k-2$.

I don’t completely understand/remember how you’re defining $cs_k^n(g)$, but since the Lie algebra cocycles I’m talking about are closely related to the Chern-Simons forms you’re describing, I’m sure your Lie 3-algebras, 5-algebras, etc. are closely related to the Lie 2-algebras, 4-algebras etc. that I’m talking about, at least in the case where $G = SU(k)$ — but in fact more generally. I’m only using $SU(k)$ as an example.

But, the interesting stuff happens when we hit Lie algebras that have characteristic classes that are not coming from the formulas you give; then we’ll get other Lie $n$-algebras.

See also the Weil algebra section of our Zoo where there is a revisionist description of the Weil algebra in terms of a Lie 2-algebra and how it played the role of forms on the universal $G$-bundle before there was one.

So, let’s comprehensively list all these facts concerning invariant polynomials, Lie algebra cocycles and Chern-Simons forms in terms of the cohomology of the qfDGCAs corresponding to $g$ $$(\wedge^\bullet ( s g) ^*, d_g)$$ and to $\mathrm{inn}(g)$, $$(\wedge^\bullet ( (s g)^* \oplus (s s g)^*), d_{\mathrm{inn}(g)}) \,.$$

What I can see so far is this:

The Lie algebra $n+1$ cocycles are precisely the elements in $$H^{n+1}(d_g) \,.$$ This should correspond to the second theorem that John mentioned.

Then, the symmetric invariant polynomials of degree $n+1$ are precisely the elements in $$H^{2n+2}(d_{\mathrm{inn}(g)})|_{(s s g)^*} \,.$$ That should essentially be the Chern-Weil theorem that John mentioned at the beginning of his comment.

Now, how does that story continue?

What is the good way to say that a given invariant polynomial admits a Chern-Simons form?

I think the point is that the polynomial $k$ must be cohomologically nontrivial (with respect to $d_{\mathrm{inn}(g)}$) as an element in $\wedge^\bullet (s s g)^*$, but not as an element of $\wedge^\bullet ( (s g)^* \oplus (s s g)^* )$.

As an element of the latter it trivializes $$k(r) = d_{\mathrm{inn}(g)}(f(t,r))$$ and the potential $f$ gives the Chern-Simons form – at least when $f$ looks like $$f(t,r) = \mu_k(t) + Q(t,r) \,,$$ where $\mu_k(t)$ is the $(2n+1)$-cocycle coming from the degree $(n+1)$ polynomial $k$, and where $Q(t,r)$ is some polynomial which is at least linear in $r$ (i.e. which vanishes when restricted to $\wedge^\bullet (s g)^*$).

So, what is the more sophisticated way of saying all this? What’s really going on when a Chern-Simons form exists for a given invariant polynomial? Does it say anything beyond that $\mu_k$ is a cocycle, hence that we have transgression here relating invariant polynomial and cocycle?

Since $\mathrm{inn}(g)$ trivializes as a Lie 2-algebra, this should mean that the $d_{\mathrm{inn}(g)}$-cohomology is trivial (but I am confused slightly about this point, related to that problem with derivation homotopies).

But if this is true, then it would mean that every invariant polynomial is cohomologically trivial when extended from $\wedge^\bullet (s s g)^*$, where it is born, to all of $\wedge^\bullet( (s g)^* \oplus (s s g)^* )$.

So then it would seem that the only further condition for a Chern-Simons form to exist is that a $Q$ exists, as above.

What does that really mean? Is that even an extra condition? What is going on here?

Whatever the answer is, it would greatly help me to see it entirely in the language of $\mathrm{inn}(g)$, if possible.