The n-Category Café

Urs wrote:

Chern-Simons Lie $(2n+1)$-algebras from degree-$(n+1)$ invariant polynomials with associated cocycle

When the invariant degree-$n+1$ polynomial $k$ induces a $(2n+1)$-cocycle $\mu_k$ as described above, there is a mixture of these two constructions.

Assume that the polynomial $k$ has the special property that… [SCARY FORMULAS OMITTED - jb]

This is for instance the case whenever the polynomial $k$ admits a Chern-Simons form.

(I was about to say something supposedly profound about if this is always possible and things like that, when I ran into a mistake I was making concerning these issues. So for the moment I won’t say anything further about this issue, except for noticing that there is at least one interesting class of polynomials $k$ where this does work.)

If I understand what you’re talking about — my eyes glaze over when I look at those scary formulas — you’re wondering if every degree-$(n+1)$ invariant polynomial on the Lie algebra $g$ comes from an element in the Lie algebra cohomology $H^{2n+1}(g, \mathbb{R})$.

Here’s a nicer way to say the same thing: you’re wondering if every element of the cohomology of $B G$ comes from an element of the cohomology of $G$ via the ‘transgression’ map

$$trans: H^{2n+1}(G,\mathbb{R}) \to H^{2n+2}(B G,\mathbb{R})$$

(Following Jim, I’ve decided to reverse the direction of my transgression map. I’ve also reindexed things to match your post.)

If that’s what you’re wondering, the answer is no!

As I tried to explain, and Jim explained much better, when $G$ is a compact Lie group the only guys in the image of $$trans: H^{2n-1}(G,\mathbb{R}) \to H^{2n}(B G,\mathbb{R})$$

are the so-called ‘primitive’ elements: linear combinations of the guys that generate $H^{2n}(B G,\mathbb{R})$ as a polynomial algebra.

For $G = SU(k)$ this means: linear combinations of the Chern classes $c_2, \dots, c_k$.

But not, for example, $c_2^2$.

So, for example, when $G = SU(k)$, we get a so-called Baez–Crans Lie 2-algebra out of the 2nd Chern class $c_2$, because this has a corresponding Chern–Simons form. In other words, it comes from an element of Lie algebra cohomology via transgression.

We also get a Lie 4-algebra from the 3rd Chern class, and a Lie 6-algebra from the 4th Chern class, and so on, because all these have corresponding Chern–Simons forms.

But, we wouldn’t get a Lie 4-algebra from $c_2^2$, because this element isn’t ‘primitive’.

If I’m failing to understand you, or you’re failing to understand me, please let me know. I could also easily be making some mistakes with all these numbers…

The nice thing is, all the primitive elements of the cohomology of $B G$ have been listed in that darn table I keep mentioning on page 11 here, so we can completely understand what’s going on, at least when $G$ is a compact simply-connected simple Lie group.

I think I have got it. (“it” means: the ${d}_{\mathrm{inn}(g)}$-cohomological formulation of the transgression scenario we were talking about) Here is how it works.

It is helpful to realize that the differential graded commutative algebra corresponding to $\mathrm{inn}(g)$ is actually free not only as a graded commutative algebra, but also as a differential algebra.

It is entriely generated from $t^a$ and $d_{\mathrm{inn}(g)} t^a$ subject to $d_{\mathrm{inn}(g)}^2=0$. One may then regard $$r^a = d t^a + \frac{1}{2}C^a{}_{bc}t^b t^c$$ as a mere abbreviation – or in fact as specifying the isomorphism from $(\mathrm{inn}(g))^*$ to the free differential graded commutative algebra over the generators $\{t^a\}$.

The point is that this means we can extend the degree -1 map $$t^a \mapsto 0$$ $$d t^a \mapsto t^a$$ to a derivation homotopy from the zero morphism on $\mathrm{inn}(g)$ to the identity on $\mathrm{inn}(g)$, i.e. to a map such that $$[d_{\mathrm{inn}(g)},\tau] = \mathrm{Id}_{\mathrm{inn}(g)} \,.$$ This $\tau$ is such that when acting on products of the free generators (and that’s where I had a stumbling block up until now) like $$t^{a_1} \wedge d t^{a_2} \wedge t^{a_3}$$ it sends one of the $d t^{a_2}$ to $t^{a_2}$, inserts some appropriate signs and sums over all permutations.

It’s not important for the following what these signs etc are in detail.

The only important point now is this:

Given an invariant symmetric polynomial, which is a $d_{\mathrm{inn}(g)}$-closed polynomial in the $r^a$ $$d_{\mathrm{inn}(g)}k(r) = 0$$ we have $$k(r) = d_{\mathrm{inn}(g)} \tau (k(r))$$ hence that Chern-Simons potential which we are looking for is $$f(t,r) := \tau(k(r)) \,.$$

To compute this by the above prescription, first replace all $r^a$ by the corresponding combination of the free generators $$d t^a + \frac{1}{2}C^a{}_{b c}t^b t^c$$ then compute $$\tau \left( k_{a_1,\cdots,a_{n+1}} (d t^{a_1} + \frac{1}{2}C^{a_1}{}_{bc}t^b t^c) \wedge \cdots \wedge (d t^{a_{n+1}} + \frac{1}{2}C^{a_{n+1}}{}_{bc}t^b t^c) \right) \,.$$

Clearly, by the above definition of $\tau$, we get all kinds of polynomials in $t$ and $r$ this way. But notice that the one polynomial which doesn’t contain any $r$s is precisely the Lie algebra cochain defined by $k$ $$\mu_k(t) = k_{a_1,\cdots,a_{n+1}} t^{a_1} \wedge (d_g t^{a_2}) \wedge \cdots (d_g t^{a_{n+1}}) \,.$$

Not in every case will this Lie algebra chochain also be a Lie algebra cocycle, i.e. be $d_g$-closed. But if it is, then the stuff computed by $\tau$ here $$\tau ( k(r) )$$ is precisely the Chern-Simons potential of the invariant polynomial $k(r)$!

It is easy to check this explicitly for $k= c_2$ the second Chern class, for instance, where it reproduces precisely the Chern-Simons potential as it appears in our discussion of the Chern-Simons Lie 3-algebra which corresponds to the Baez-Crans-type String Lie 2-algebra.

Hey guys,
Let’s go over the basics more thoroughly.
For simplicity, let’s consider G = U(n)
Then H again being cohomology
H(G) is a Hopf algebra generated by x_i of
degree 2i-1
which can be taken to be primitive.
In this example, primitive implies transgressive
so transgression is defined on the vector subspace with basis the x_i

transgression is well defined only modulo products
i.e transgression: Prim H(G) –> Indecomposables of H(BG)

in particular,
transgression(x_i) is represented by c_i
the i-th Chern class and H(BG) is the polynomial alg generated by the c_i, i lesseq n

any principal G-bundle G –> P –> B
is classified by a homotopy class of maps B –> BG
and it’s characteritic classes are the pull backs of classes in H(BG)

Chern-Simons forms are closed if they are
of the same degree as the top dim of B

Here is another comment on how the computations within the differential graded algebra version of the Lie 2-algebra $\mathrm{inn}(g)$ that I kept talking about should be directly related to the topological aspects you emphasized:

I guess the important point is that we may think of the universal $G$-bundle $$G \to E G \to B G$$ as the “exact” sequence of categories $$\mathrm{Disc}(G) \to \mathrm{INN}(G) \to \Sigma(G)$$

Here $\mathrm{Disc}(G)$ denotes the category which has just the elements of the group $G$ as objects and only identity morphisms on these, $\mathrm{INN}(G)$ is the 2-group with $G$ as its space of objects and $G \times G$ as its space of morphisms $g \stackrel{h}{\to } h g$ and $\Sigma G$ is the category with a single object and elements of $G$ as morphisms on that.

By taking the geometric realization of this sequence of categories we reobtain the universal $G$ bundle $$(|\mathrm{Disc}(G)| \to |\mathrm{INN}(G)| \to |\Sigma G|) \simeq ( G \to E G \to B G) \,.$$

This picture seems to explain completely how the lore of characteristic classes in terms of differential form on $G$ bundles that you recalled and emphasized can be rediscovered in terms of the cohomology of the differential algebra which describes $$\mathrm{inn}(g) := \mathrm{Lie}(\mathrm{INN}(G)) \,.$$

Clearly, the existence of the above sequence translates into the sequence $$(s g)^* \to ( (sg)^* \oplus (s s g)^*) \to (s s g)^* \,.$$ On the frist two items we have a Lie $n$-algebra strcuture, witnessed by the existence of a nilpotent differential on the freely generated graded commutative algebra:

there is $d_g$ acting on $\wedge^\bullet (s g)^*$, coming from the Lie group structure on $G$, and there is $d_{\mathrm{inn}(g)}$ acting on $\wedge^\bullet ( (sg)^* \oplus (s s g)^*)$, coming from the Lie 2-group structure on $\mathrm{INN}(G)$.

Since $\Sigma(G)$ is not in general a 2-group (unless $G$ is abelian), we don’t have a differential on $\wedge^\bullet (s s g)^*$ alone.

Now, we can translate the description of transgression of characterictic classes and Lie algebra cocycles which Jim Stasheff kindly recalled in our other thread in terms of this DGCA language (or, I guess: in Weil-algebra language, by means of our “revisionist point of view” which identitfies the Weil algebra with the differential graded commutative algebra $(\mathrm{inn}(g))^* = (\wedge^\bullet( (sg)^* \oplus (s s g)^*),d_{\mathrm{inn}(g)})$).

First, we want a closed differential form $w$ on the fiber $G$. That corresponds to a $d_g$-closed element in $\wedge^\bullet (s g)^*$.

Then we extend this to a form $v$ on all of $E G$. This means we extend it to an element in $\wedge^\bullet ( (s g)^* \oplus (s s g)^*)$ such that its closure there, namely $$d_{\mathrm{inn}(g)} v$$ is a form $u$ pulled back from base space. But this means, by the above dictionary, that $u$ is an element of $$\wedge^\bullet (s s g)^* \,.$$

So, in summary, the transgression between an invariant $n+1$ polynomial and $(2n+1)$-cocycle is, in terms of the DGCAs which we obtain by differentiating $\mathrm{Disc}(G) \to \mathrm{INN}(G) \to \Sigma G$ the following situation:

given a $d_g$-closed element $w$ in $\wedge^\bullet (s g)^*$ and a $d_{\mathrm{inn}(g)}$-closed element $u$ in $\wedge^\bullet (s s g)^*$, find an element $$v \in \wedge^\bullet( (s g)^* \oplus (s s g)^* )$$ such that

a) $v$ restricted to $\wedge^\bullet (s g)^*$ equals $w$

b ) $d_{\mathrm{inn}(g)} v = u$

That’s it.

And what I was pointing out is that there is a systematic way to compute this as follows: start with $u \in \wedge^\bullet( s s g)^*$ and apply the degree -1 map $$\tau \in \mathrm{End}(\wedge^\bullet ( (s g)^* \oplus (s s g)^* ))\,.$$

Then,

- either $(\tau u)|_{\wedge^\bullet (s g)^*}$ is $d_g$-closed - then it is the cocycle related to $u$ by transgression

- or $(\tau u)|_{\wedge^\bullet (s g)^*}$ is not $d_g$-closed, then $u$ is not related by transgression to any cocycle.

With the broader perspective on higher Chern-Simons Lie algebras finally understood, we might be in better position for understanding the broader context in which to think of the supergravity super Lie 3-algebra $$\mathrm{sugra}(10,1)$$ (discussed at length here and here).

First of all, it is clear that this is a Lie $n$-algebra of super Baez-Crans type: it comes precisely from one $d_g$-closed element $$\bar \psi \wedge \Gamma^{a b}\psi \wedge e_a \wedge e_b$$ in $\wedge^\bullet (s g)^*$, for $g$ the super Poincaré Lie algebra.

(Important point here: since $g$ itself is now $\mathbb{Z}_2$-graded, $\wedge^\bullet (s g)^*$ is bigraded and we have for instace $\psi^\alpha \wedge \psi^\beta = + \psi^\beta \wedge \psi^\alpha$.)

So, given the above, this makes us want to check if this cocycle transgresses, i.e. of there is a super characteristic class $k$ and a Chern-Simons potential such that $\mathrm{sugra}(10,1)$ fits into an exact sequence $$0 \to \mathrm{sugra}(10,1) \to \mathrm{cs}_k(g) \to \mathrm{ch}_k(g) \to 0 \,.$$

This requires finding out if the above cocycle transgresses to a super characteristic class.

That would be potentially interesting. This characteristic class should be a polynomial in the supercurvatures, which are the ordinary $so(s,1)$ curvature $$R^{ab}$$ the (super-corrected) torsion $$R^a$$ and the spinorial curvature $$\rho := D \psi$$ which is nothing but the image of our spinors under the Dirac operator corresponding to the given $so(s,1)$-connection.

I am guessing that with the cocycle even-graded, the characteristic class will have to be odd graded, which probably means we have to throw in one spinor and instead of the 2-form $\rho$ use the 3-form $$\bar \psi \rho = \bar \psi D \psi \,.$$

So maybe the characteristic class here should look like $$R_{a}\wedge R_b\wedge\bar \psi \wedge\Gamma^{ab} D \psi$$ or something.

I need to find

D. Leites and D. B. Fuchs, Cohomology of Lie superalgebras, C. R. Acad. Bulg. Sci. 37 (1984), 1595-1596.

Can anyone provide me with helpful literature?

Or, better yet, does anyone even know the answer to: does the super $iso(10,1)$-cocycle $\bar \psi \Gamma^{ab} \wedge \psi \wedge e_a \wedge e_b$ transgress, and if so, to which characteristic class?

Before I (of all people) attempt to say something about characterisitc classes for
Principal 2-group bundles, what are the latter?

To me, the naive definition is (at least in the crossed module case) with the 2-group
expressed as T:H –> G,
a principal (T:H –> G) bundle consists of
a compatible pair of principal bundles
P–> B a principal H-bundle and Q–>B
a principal G-bundle with an equivariant map
P –> Q over B.
But then whe have the characterisitc classes c_P and c_Q in H*(B)

Indeed, this works universally for
EH and EG
where now we can compare over BH –> BG.

Shoudl I be trying something more subtle?

jim