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May 19, 2007

Chern Lie (2n+1)-Algebras

Posted by Urs Schreiber

I mentioned Lie (2n+1)(2n+1)-algebras coming from invariant symmetric polynomials kk of degree n+1n+1 on some Lie algebra gg in Zoo of Lie nn-Algebras.

Since it is a really simple and short computation, I want to give here all the details needed to understand it.

This is the picture I want to establish:

This is really a weakly (up to homotopy) exact sequence of Lie (2n+1)(2n+1)-algbras 0g μ kcs k(g)ch k(g)0 0 \to g_{\mu_k} \to \mathrm{cs}_k(g) \to \mathrm{ch}_k(g) \to 0 whenever kk induces a (2n+1)(2n+1)-cocycle μ k\mu_k.

I’ll work here entirely in terms of differential graded commutative algebras, which are free as graded commutative algebras. Alternatively I could give the codifferential coalgebra picture, which has the advantage that it is practical to write everything down without first choosing a basis. But here I won’t. This means I’ll chose a basis for displaying my formulas. Those who don’t like this can easily translate back to the coalgebra picture.


Basic machinery

So, fix some Lie algebra gg. Write (sg)(s g) for a copy of the underlying vector space, but considered as being in degree 1, and write (ssg)(s s g) for yet another copy of that vector space, but considered now in degree 2. And so on.

Choose a basis {X a}\{X_a\} of gg and let {t a}\{t^a\} be the corresponding dual basis of (sg) *(s g)^*. Let C a bcC^a{}_{bc} be the structure constants of gg in that basis.

Then, on the free graded commutative algebra (sg) * \wedge^\bullet (s g)^* the following defines a graded differential dd of degree +1 such that d 2=0d^2 = 0: d gt a=12C a bct bt c. d_g t^a = - \frac{1}{2} C^a{}_{bc}t^b \wedge t^c \,. The differential algebra g *:=( (sg) *,d g) g^* := (\wedge^\bullet (s g)^*, d_g) is the Lie algebra gg, in its dual incarnation.

Next, consider the free differential graded algebra generated from two copies of g *g^*, in degree 1 and 2: ( (sg) *(ssg) *). (\wedge^\bullet ( s g)^* \oplus (s s g)^* ) \,. Write {r a}\{r^a\} for the dual basis of {X a}\{X_a\}, but regarded as being in degree 2. Then d inn(g)t a=12C a bct bt c+r a d_{\mathrm{inn}(g)} t^a = - \frac{1}{2} C^a{}_{bc}t^b \wedge t^c + r^a and d inn(g)r a=C a bct br c d_{\mathrm{inn}(g)} r^a = C^a{}_{bc}t^b \wedge r^c defines again a differential of degree +1 and squaring to 0. The differential algebra ( (sg) *(ssg) *,d inn(g)) (\wedge^\bullet ( s g)^* \oplus (s s g)^* , d_{\mathrm{inn}(g)}) is the (dual incarnation of the) Lie 2-algebra called inn(g). \mathrm{inn}(g) \,.


Cocycles and Invariant Polynomials in inn(g)\mathrm{inn}(g)-cohomology

It will be very convenient to formulate the concept of Lie algebra cohomology in terms of the above differential algebras. (The following is my observation. If it is well known, which it should be in as far as it is right, I am not aware of the relevant literature.)

- A coycle μ\mu of degree n+1n+1 on the Lie algebra gg is nothing but a d gd_g-closed polynomial of degree n+1n+1 in ((sg) *)\wedge^\bullet ( (s g)^* ): d gμ(t)=0. d_g \mu(t) = 0 \,.

- A symmetric invariant polynomial kk of degree n+1n+1 on gg is nothing but a d inn(g)d_{\mathrm{inn}(g)}-closed polynomial in (ssg) *\wedge\bullet ( s s g)^*: d inn(g)k(r)=0. d_{\mathrm{inn}(g)} k(r) = 0 \,.

Notice that, in components, μ\mu is necessarily given by an antisymmetric and kk by a symmetric tensor, since the t at^a are odd graded while the r ar^a are even graded.

- Given an invariant polynomial kk as above, it may happen that k a 0,a 1a nt a 0(dt a 1)(dt a n) k_{a_0, a_1 \cdots a_n} t^{a_0} \wedge (d t^{a_1}) \wedge \cdots \wedge (d t^{a_n}) is a cocycle. If so, I’ll call this cocycle μ k(t). \mu_k(t) \,.


Baez-Crans Lie nn-algebras from (n+1)(n+1)-cocycles

Given an (n+1)(n+1)-cocycle μ\mu on gg, it is easy to extended the differential algebra g *g^* to one whose differential d g μd_{g_\mu} is defined on ((sg) *(s n) *) \wedge^\bullet ( (s g)^* \oplus (s^{n} \mathbb{R})^* ) by d g μt a=12C a bct bt c d_{g_\mu} t^a = - \frac{1}{2} C^a{}_{bc}t^b \wedge t^c as before and d g μb=μ(t), d_{g_\mu} b = - \mu(t) \,, where {b}\{b\} is the canonical basis of (s n) *(s^n \mathbb{R})^*, i.e nothing but a linear form on \mathbb{R}, regarded as being in degree nn.

Clearly, (d g μ) 2=0(d_{g_\mu})^2 = 0 by the defining property of μ\mu.

The resulting differential algebra is the (dual incarnation of the ) Baez-Crans Lie nn-algebra g μ. g_\mu \,.


Chern Lie (2n+1)(2n+1)-algebras from degree (n+1)(n+1)-invariant polynomials

Quite analogously, we can construct a nilpotent differential from any invariant polynomial. All we have to do is to allow for the existence of even-graded elements. So we essentially repeat the above construction, but building on inn(g)\mathrm{inn}(g) instead of on gg itself.

So, given a degree (n+1)(n+1)-invariant polynomial kk on gg, it is easy to extended the differential algebra inn(g) *\mathrm{inn}(g)^* to one whose differential d ch k(g)d_{\mathrm{ch}_k(g)} is defined on ((sg) *(ssg) *(s 2n+1) *) \wedge^\bullet ( (s g)^* \oplus (s s g)^* \oplus (s^{2n+1} \mathbb{R})^* ) by d ch k(g)t a=12C a bct bt c+r a d_{\mathrm{ch}_k(g)} t^a = - \frac{1}{2} C^a{}_{bc}t^b \wedge t^c + r^a d ch k(g)r a=C a bct br c d_{\mathrm{ch}_k(g)} r^a = C^a{}_{bc}t^b \wedge r^c as before and d ch k(g)c=k(r), d_{\mathrm{ch}_k(g)} c = k(r) \,, where {c}\{c\} is the canonical basis of (s 2n+1) *(s^{2n+1} \mathbb{R})^*, i.e nothing but a linear form on \mathbb{R}, regarded as being in degree 2n+12n+1.

Clearly, (d ch k(g)) 2=0(d_{\mathrm{ch}_k(g)})^2 = 0 by the defining property of kk.

The resulting differential algebra might be called the (dual incarnation of the ) Chern Lie nn-algebra ch k(g). \mathrm{ch}_k(g) \,.


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

When the invariant degree n+1n+1 polynomial kk induces a 2n+12n+1-cocycle μ k\mu_k as described above, there is a mixture of these two constructions.

Assume that the polynomial kk has the special property that there is a polynomial Q(t,r)=Q a(t,r)r a Q(t,r) = Q_a(t,r)r^a (meaning it’s a polynomial in the tts and in the rrs which is at least linear in the rrs) such that k(r)=d inn(g)(μ k(t)+Q(t,r)). k(r) = d_{\mathrm{inn}(g)} ( \mu_k(t) + Q(t,r) ) \,. This is for instance the case whenever the polynomial kk 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 kk where this does work.)

Given the above situation, it is easy to extended the differential algebra ch k(g) *\mathrm{ch}_k(g)^* to one whose differential d cs k(g)d_{\mathrm{cs}_k(g)} is defined on ((sg) *(ssg) *(s 2) *(s 2n+1) *) \wedge^\bullet ( (s g)^* \oplus (s s g)^* \oplus (s^{2} \mathbb{R})^*\oplus (s^{2n+1} \mathbb{R})^* ) by d cs k(g)t a=12C a bct bt c+r a d_{\mathrm{cs}_k(g)} t^a = - \frac{1}{2} C^a{}_{bc}t^b \wedge t^c + r^a d cs k(g)r a=C a bct br c d_{\mathrm{cs}_k(g)} r^a = C^a{}_{bc}t^b \wedge r^c d ch k(g)c=k(r) d_{\mathrm{ch}_k(g)} c = k(r) as before and db=(μ(t)+Q(t,r))+c. d b = - (\mu(t) + Q(t,r)) + c \,.

Clearly, (d g μ) 2=0(d_{g_\mu})^2 = 0 by the defining property of kk and Q.

The resulting differential algebra might be called the (dual incarnation of the ) Chern-Simons Lie (2n+1)(2n+1)-algebra cs k(g). \mathrm{cs}_k(g) \,.


The canonical morphisms between these Lie nn-algebras

We the obvious inclusion ((sg) *(ssg) *(s 2n+1) * ((sg) *(ssg) *(s 2) *(s 2n+1) * \wedge^\bullet ( (s g)^* \oplus (s s g)^* \oplus (s^{2n+1} \mathbb{R})^* \to \wedge^\bullet ( (s g)^* \oplus (s s g)^* \oplus (s^{2} \mathbb{R})^*\oplus (s^{2n+1} \mathbb{R})^* and the obvious surjection ((sg) *(ssg) *(s 2n) *(s 2n+1) * ((sg) *(ssg) *(s 2n) *). \wedge^\bullet ( (s g)^* \oplus (s s g)^* \oplus (s^{2n} \mathbb{R})^*\oplus (s^{2n+1} \mathbb{R})^* \to \wedge^\bullet ( (s g)^* \oplus (s s g)^* \oplus (s^{2n} \mathbb{R})^*) \,. Its is easy to check that both these maps commute with the respective differentials and that their composite is homotopic to the zero map.

This means that we have an exact sequence of Lie (2n+1)(2n+1)-algebras 0g μ kcs k(g)ch k(g)0. 0 \to g_{\mu_k} \to \mathrm{cs}_k(g) \to \mathrm{ch}_k(g) \to 0 \,.

Posted at May 19, 2007 2:52 PM UTC

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18 Comments & 8 Trackbacks

Re: Chern Lie (2n+1)-Algebras

Urs wrote:

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

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

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

This is for instance the case whenever the polynomial kk 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 kk 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)(n+1) invariant polynomial on the Lie algebra gg comes from an element in the Lie algebra cohomology H 2n+1(g,)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 BGB G comes from an element of the cohomology of GG via the ‘transgression’ map

trans:H 2n+1(G,)H 2n+2(BG,)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 GG is a compact Lie group the only guys in the image of trans:H 2n1(G,)H 2n(BG,)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(BG,)H^{2n}(B G,\mathbb{R}) as a polynomial algebra.

For G=SU(k)G = SU(k) this means: linear combinations of the Chern classes c 2,,c kc_2, \dots, c_k.

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

So, for example, when G=SU(k)G = SU(k), we get a so-called Baez–Crans Lie 2-algebra out of the 2nd Chern class c 2c_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 2c_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 BGB 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 GG is a compact simply-connected simple Lie group.

Posted by: John Baez on May 20, 2007 12:34 AM | Permalink | Reply to this

Re: Chern Lie (2n+1)-Algebras

Thanks. I hope I am not making the impression that I am not at all listening to what you and Jim keep saying. Quite the opposite.

Probably – that’s at least the impression I get – what I am talking about in that paragraph which you quoted is what the transgression map looks like at the level of the cohomology of inn(g)\mathrm{inn}(g).

Whether or not, I’ll need to understand it eventually at that level, because at that level the Lie nn-algebras are built (or dually in the coalgebra, if one prefers, but doesn’t really make much of a difference except for notation).

So, there are two aspects to this, then:

a) is the cohomology of d inn(g)d_{\mathrm{inn}(g)}-trivial?

I thought it is, since I thought that inn(g)\mathrm{inn}(g) is trivializable as a Lie 2-algebra, which means that there is a “homotopy operator” τ\tau of degree -1 such that [d inn(g),τ]=Id inn(g). [d_{\mathrm{inn}(g)},\tau] = \mathrm{Id}_{\mathrm{inn}(g)} \,. Maybe I am making a mistake here. But if not, this means that every invariant polynomial is d inn(g)d_{\mathrm{inn}(g)}-exact. k(r)=d inn(g)f(t,r)k(r) = d_{\mathrm{inn}(g)} f(t,r).

b) but that’s not sufficient for it to admit a Chern-Simons construction. For that to happen, the potential f(t,r)f(t,r) needs to restrict, when we set r0r\mapsto 0, to the cocycle μ k(t)\mu_k(t).

I guess that’s where the transgression thing comes in. As Jim seems to have mentioned, the transgression is essentially mediated by the Chern-Simons form itself.

And indeed, in precisely the cases that you mention, namely where kk is one of the Chern-classes, I can explicitly write down the right potential f(t,r)f(t,r): it is nothing but the well known Chern-Simons form, with AA replaced by tt and F AF_A replaced by rr.

So, it all seems to make sense, and I am looking forward to when I am sure enough about this to be able to take that table which you keep pointing me to and read off all the Chern-Simons Lie (2n+1)(2n+1)-algebras that can be built using the information summarized there.

Only thing I should try to get clarified first is how exactly the transgression map looks like in the world of the qfDGCA of inn(g)\mathrm{inn}(g).

Posted by: urs on May 20, 2007 1:08 AM | Permalink | Reply to this

Re: Chern Lie (2n+1)-Algebras

I think I have got it. (“it” means: the d inn(g){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 inn(g)\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 at^a and d inn(g)t ad_{\mathrm{inn}(g)} t^a subject to d inn(g) 2=0d_{\mathrm{inn}(g)}^2=0. One may then regard r a=dt a+12C a bct bt c 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 (inn(g)) *(\mathrm{inn}(g))^* to the free differential graded commutative algebra over the generators {t a}\{t^a\}.

The point is that this means we can extend the degree -1 map t a0 t^a \mapsto 0 dt at a d t^a \mapsto t^a to a derivation homotopy from the zero morphism on inn(g)\mathrm{inn}(g) to the identity on inn(g)\mathrm{inn}(g), i.e. to a map such that [d inn(g),τ]=Id inn(g). [d_{\mathrm{inn}(g)},\tau] = \mathrm{Id}_{\mathrm{inn}(g)} \,. This τ\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 1dt a 2t a 3 t^{a_1} \wedge d t^{a_2} \wedge t^{a_3} it sends one of the dt a 2d t^{a_2} to t a 2t^{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 inn(g)d_{\mathrm{inn}(g)}-closed polynomial in the r ar^a d inn(g)k(r)=0 d_{\mathrm{inn}(g)}k(r) = 0 we have k(r)=d inn(g)τ(k(r)) k(r) = d_{\mathrm{inn}(g)} \tau (k(r)) hence that Chern-Simons potential which we are looking for is f(t,r):=τ(k(r)). f(t,r) := \tau(k(r)) \,.

To compute this by the above prescription, first replace all r ar^a by the corresponding combination of the free generators dt a+12C a bct bt c d t^a + \frac{1}{2}C^a{}_{b c}t^b t^c then compute τ(k a 1,,a n+1(dt a 1+12C a 1 bct bt c)(dt a n+1+12C a n+1 bct bt c)). \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 τ\tau, we get all kinds of polynomials in tt and rr this way. But notice that the one polynomial which doesn’t contain any rrs is precisely the Lie algebra cochain defined by kk μ k(t)=k a 1,,a n+1t a 1(d gt a 2)(d gt a n+1). \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 gd_g-closed. But if it is, then the stuff computed by τ\tau here τ(k(r)) \tau ( k(r) ) is precisely the Chern-Simons potential of the invariant polynomial k(r)k(r)!

It is easy to check this explicitly for k=c 2k= 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.

Posted by: urs on May 20, 2007 12:36 PM | Permalink | Reply to this

Re: Chern Lie (2n+1)-Algebras

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

Posted by: jim stasheff on May 20, 2007 4:21 PM | Permalink | Reply to this

Re: Chern Lie (2n+1)-Algebras

Here is another comment on how the computations within the differential graded algebra version of the Lie 2-algebra inn(g)\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 GG-bundle GEGBG G \to E G \to B G as the “exact” sequence of categories Disc(G)INN(G)Σ(G) \mathrm{Disc}(G) \to \mathrm{INN}(G) \to \Sigma(G)

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

By taking the geometric realization of this sequence of categories we reobtain the universal GG bundle (|Disc(G)||INN(G)||ΣG|)(GEGBG). (|\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 GG bundles that you recalled and emphasized can be rediscovered in terms of the cohomology of the differential algebra which describes inn(g):=Lie(INN(G)). \mathrm{inn}(g) := \mathrm{Lie}(\mathrm{INN}(G)) \,.

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

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

Since Σ(G)\Sigma(G) is not in general a 2-group (unless GG is abelian), we don’t have a differential on (ssg) *\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 (inn(g)) *=( ((sg) *(ssg) *),d inn(g))(\mathrm{inn}(g))^* = (\wedge^\bullet( (sg)^* \oplus (s s g)^*),d_{\mathrm{inn}(g)})).

First, we want a closed differential form ww on the fiber GG. That corresponds to a d gd_g-closed element in (sg) *\wedge^\bullet (s g)^*.

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

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

given a d gd_g-closed element ww in (sg) *\wedge^\bullet (s g)^* and a d inn(g)d_{\mathrm{inn}(g)}-closed element uu in (ssg) *\wedge^\bullet (s s g)^*, find an element v ((sg) *(ssg) *) v \in \wedge^\bullet( (s g)^* \oplus (s s g)^* ) such that

a) vv restricted to (sg) *\wedge^\bullet (s g)^* equals ww

b ) d inn(g)v=ud_{\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 (ssg) *u \in \wedge^\bullet( s s g)^* and apply the degree -1 map τEnd( ((sg) *(ssg) *)). \tau \in \mathrm{End}(\wedge^\bullet ( (s g)^* \oplus (s s g)^* ))\,.

Then,

- either (τu)| (sg) *(\tau u)|_{\wedge^\bullet (s g)^*} is d gd_g-closed - then it is the cocycle related to uu by transgression

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

Posted by: urs on May 20, 2007 9:30 PM | Permalink | Reply to this

Super Chern Lie (2n+1)-algebras?

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 sugra(10,1) \mathrm{sugra}(10,1) (discussed at length here and here).

First of all, it is clear that this is a Lie nn-algebra of super Baez-Crans type: it comes precisely from one d gd_g-closed element ψ¯Γ abψe ae b \bar \psi \wedge \Gamma^{a b}\psi \wedge e_a \wedge e_b in (sg) *\wedge^\bullet (s g)^*, for gg the super Poincaré Lie algebra.

(Important point here: since gg itself is now 2\mathbb{Z}_2-graded, (sg) *\wedge^\bullet (s g)^* is bigraded and we have for instace ψ αψ β=+ψ βψ α\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 kk and a Chern-Simons potential such that sugra(10,1)\mathrm{sugra}(10,1) fits into an exact sequence 0sugra(10,1)cs k(g)ch k(g)0. 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)so(s,1) curvature R ab R^{ab} the (super-corrected) torsion R a R^a and the spinorial curvature ρ:=Dψ \rho := D \psi which is nothing but the image of our spinors under the Dirac operator corresponding to the given so(s,1)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 ρ\rho use the 3-form ψ¯ρ=ψ¯Dψ. \bar \psi \rho = \bar \psi D \psi \,.

So maybe the characteristic class here should look like R aR bψ¯Γ abDψ 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)iso(10,1)-cocycle ψ¯Γ abψe ae b\bar \psi \Gamma^{ab} \wedge \psi \wedge e_a \wedge e_b transgress, and if so, to which characteristic class?

Posted by: urs on May 21, 2007 5:55 PM | Permalink | Reply to this

Re: Super Chern Lie (2n+1)-algebras?

Hisham Sati was so kind to point me to this reference, which has some details on the Chern-Simons 3-form for super-Poincaré in 10-dimensions:

Bonora, Bregola, D’Auria, Fre, LEchner, Pasti, Pesando, Raciti, Riva, Tonin, Zanon: Some remarks on the supersymmetrization of the Lorentz Chern-Simons form in D=10D=10 N=1N=1 supergravity theories (pdf).

That doesn’t feature explicitly that 4-cocycle we were talking about. But somehow this must be related.

The mixed term in the super Chern-Simons form ψ¯Γ aψe a \bar \psi \wedge \Gamma^a \psi \wedge e_a (ψ\psi the spinor-valued 1-form, ee the Vielbein 1-form) looks like it should come from our 4-cocycle in eleven dimensions ψ¯Γ abψe ae b \bar \psi \wedge \Gamma^{ab} \psi \wedge e_a \wedge e_b by “compactifying one dimension”, also known as integrating this over an S 1S^1-fiber.

Posted by: urs on May 25, 2007 4:57 PM | Permalink | Reply to this

Re: Chern Lie (2n+1)-Algebras

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

Posted by: jim stasheff on May 22, 2007 12:54 AM | Permalink | Reply to this

Re: Chern Lie (2n+1)-Algebras

There have been some reactions to this question behind the scenes. I’d have questions concerning these reactions, but for the moment I just give my reply here out in the open:

Should I be trying something more subtle?

In general, yes!

What you describe are principal 2-bundles for a strict 2-group which are such that each fiber looks like a strict torsor (“over a point”) for the strict 2-group.

These are example of principal 2-bundles, but very restricted ones. Most of the really interesting examples are not obtained this way.

Just take the simplest case to see this: let the 2-group G (2)G_{(2)} be shifted U(1)U(1):

G (2):=ΣU(1):=(U(1)1). G_{(2)} := \Sigma U(1) := (U(1) \to 1) \,. Then principal G (2)G_{(2)}-2-bundles should be categorified circle bundles, which should be equivalent to abelian gerbes.

But by your prescription all you’d get from this is just an ordinary principal circle bundle! This may be regarded as trivializing a trivializable abelian gerbe. But certainly we want more general gerbes, too.

In fact, the situation is even worse than that. For the total space P\mathbf{P} of your 2-bundle to really be a category over the discrete category Disc(B)\mathrm{Disc}(B) of base space BB π:PDisc(P) \pi : \mathbf{P} \to \mathrm{Disc}(P) you need to specify the identity morphisms in P\mathbf{P} (otherwise it’s not a category). But that means specifying a morphism from your bundle of objects QQ to your bundle of morphisms P×QP \times Q.

For the case that G (2)=(U(1)1)G_{(2)} = (U(1) \to 1) this means choosing a section of PP. Hence 2-circle bundles of the type you mentioned are not just ordinary circle bundles, but even trivial ordinary circle bundles.

When I frist ran into precisely this issue, it made me do two things:

a) think about how to weaken the notion of (U(1)1)(U(1)\to 1)-torsor (“over a point”) sufficiently such as to be able to obtain the total space of a nontrivial (U(1)1)(U(1)\to 1)-2-bundle. My thoughts on this I had once posted as How many circles are there in the world?

b) try to do away with the need for explicit total spaces of nn-bundles. Total spaces for nn-bundles are a nuisance. And according to this maxim maybe this should tell us something.

An overview of my ideas on conceiving smooth principal 22-bundles as global objects (i.e. without resorting to their cocycle description with respect to a choice of local trivialization) but without explicitly presenting their total 2-spaces, I recently gave in The first edge of the cube.

A detailed explicit discussion about how this does solve the above problem in the case of G (2)=(U(1)1)G_{(2)} = (U(1)\to 1) is here.

As a slogan, the idea of this approach is:

Don’t read an nn-bundle as a fibration π:PDisc(B) \pi : \mathbf{P} \to \mathrm{Disc}(B) and try to put a smooth structure on P\mathbf{P}, but read it the other way round as a fiber-assigning functor π 1:Disc(B)G (n)Tor \pi^{-1} : \mathrm{Disc}(B) \to G_{(n)}\mathrm{Tor} and figure out what it means to put a smooth structure on such a functor.

I should add that after talking to Toby Bartels about it, I finally began to understand how he manages, in his thesis to “get” a sensible total 2-space for a 2-bundle from any given cocycle data. As best as I can tell this procedure at least guarantees that for any given smooth cocycle data for a 2-bundle a smooth total 2-space does exist (at least under certain assumptions). I still don’t have enough practical experience with Toby’s approach to really say how much that allows us to do in practice with the total spaces defined this way.

But it does assure us that they exist!

Posted by: urs on May 23, 2007 11:09 AM | Permalink | Reply to this

Re: Chern Lie (2n+1)-Algebras

Dispensing with total spaces would certainly
make ‘interpreting’ gerbes much easier,
but where does that put the physics — globally??

Posted by: jim stasheff on May 26, 2007 1:49 PM | Permalink | Reply to this

Re: Chern Lie (2n+1)-Algebras

Jim wrote:

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

I won’t answer that, since you probably know by now, and my student Toby Bartels wrote a big fat thesis on precisely this topic… probably too big and fat for easy reading, but certainly quite precise.

But about the former: Danny Stevenson and I are writing a paper in which we construct a classifying space for principal GG-2-bundles for any topological 2-group GG. This should be a nice foundation for studying characteristic classes of such 2-bundles.

Computing them will take some work… and at least over \mathbb{R}, you and Urs can do it a lot faster than I ever would, simply by taking the Chern–Weil machinery and boosting it up from Lie algebras to Lie 2-algebras.

Some sort of prelimary writeup of this classifying space business should appear pretty soon. At the latest, by early August, I can show you transparencies of some talks I’m giving this summer.

Posted by: John Baez on May 26, 2007 1:16 AM | Permalink | Reply to this

Re: Chern Lie (2n+1)-Algebras

Toby,
Could you extract the definition of
a principal 2-group bundle and share it?

jim

Posted by: jim stasheff on May 26, 2007 1:39 PM | Permalink | Reply to this

Principal 2-Bundles

Could you extract the definition of a principal 2-group bundle and share it?

A principal 2-bundle is a category PP together with a surjective functor π:PDisc(X), \pi : P \to \mathrm{Disc}(X) \,, where Disc(X)\mathrm{Disc}(X) is the discrete category over a space XX (i.e. having only identity morphisms) such that a 2-group GG acts principally on that.

Toby defines what it means for a 2-group to act principally on a 2-bundle by saying that locally the bundle looks like base space times the 2-group P| UDisc(U)×G (2) P|_{U} \simeq \mathrm{Disc}(U) \times G_{(2)} and that the transitions between local trivializations are given by acting with G (2)G_{(2)}.

(This way, he can in the same manner also treat all associated 2-bundles.)

Alternatively, one could use the definition that the action P×G (2)PP \times G_{(2)} \to P is principal if the canonical morphism P×G (2)P× XP P \times G_{(2)} \to P \times_X P is an equivalence. That’s what Igor Bakovic uses .

In any case, one finds that the fibers of a principal 2-bundle look like torsors for the 2-group G (2)G_{(2)}.

One can easily check that a strict 2-torsor for a strict 2-group is the same as a category internal to the category of ordinary torsors.

So a strict torsor for the strict 2-group (HG)(H \to G) is

an HH-torsor T 1T_1 (“torsor of morphisms”)

and

a GG-torsor T 0T_0 (“torsor of objects”)

and a morphism

T 1T 0 T_1 \to T_0

and so on. Hence one might be tempted to look at those 2-bundles whose total space is given by a principal HH-bundle with a morphism to a principal GG-bundle.

But that’s too strict to capture the general case.

But a slight variant of this does work: Danny Stevenson has a formulaiton of nonabelian principal 2-bundles which live not over the discrete category of base space XX, but over the pair groupoid of XX.

This can be understood as the nonabelian bundle gerbe version of 2-bundles. And then one does find that the 2-bundle appears as an HH-bundle over morphisms in the pair groupoid, and a GG-bundle over XX itself.

Posted by: urs on May 28, 2007 10:21 AM | Permalink | Reply to this

Re: Chern Lie (2n+1)-Algebras

One representative of the ordinary classifying space BG is the ‘spatial’ realization of the nerve of the category which I think you guys call ΣG.

So for 2-groups, is it more subtle than realization of the nerve of the 2-category for which, I seem to recall, there are various definitions, but hopefully the realizations are homotopy equivalent.

Posted by: jim stasheff on May 26, 2007 1:44 PM | Permalink | Reply to this

Re: Chern Lie (2n+1)-Algebras

I’ve just rediscovered Jurco’s paper. Will have to absorb that, though any help welcome.

Posted by: jim stasheff on May 27, 2007 8:23 PM | Permalink | Reply to this

Re: Chern Lie (2n+1)-Algebras

The cohomology of dg Lie algebras has been around for some time implicitly though I am aware of no computations. In the notation Urs and I have been using for some time and for ground field coefficients, it’s just the homology of the complex (sgssh) \wedge^\bullet(s g \oplus ss h)

Consider the case in which the internal differential d:hg=0d:h \to g = 0. Then the cohomology splits. The piece (sg)\wedge^\bullet(s g) is the usual Lie algebra cohomology of gg while the other piece is the ‘odd’ version, for which I don’t know the computation but it should be straight forward.

Now if d0d\neq 0, we can look at the associated spectral sequence in which the E 1E_1 page is as above. The d 1d_1 differential is induced by the internal dd and should be quite computable.

Then we can attack the problem of transgression in the Weil algebra aka inn and hence get characteristic classes.

Posted by: Jim Stasheff on May 28, 2007 10:30 AM | Permalink | Reply to this

Re: Chern Lie (2n+1)-Algebras

I made a gross error
forgetting that \hh is a \gg module
so the piece \bigwedge s\gg is as I stated
giving the standard Lie alg cohomology of \gg with coefficeints in R or C
BUT
but the other piece
gives the standard Lie alg cohomology of \gg with coefficeints in \hh
or rather in \bigwedge s\hh

can’t recall seeing the latter computed

will ask around

Posted by: jim stasheff on May 28, 2007 1:46 PM | Permalink | Reply to this

Re: Chern Lie (2n+1)-Algebras

I have started writing a more coherent summary on how Lie algebra cocycles, invariant polynomials and Chern-Simons elements appear in terms of the differential algebra corresponding to the Lie 2-algebra inn(g)\mathrm{inn}(g), and how that helps to see how each transgressive n+1n+1 cocycle gives rise to an exact sequence of Lie (2n+1)(2n+1)-algebras:

Lie algebra cohomology and inn(g)\mathrm{inn}(g)

In summary, the situation is the following:

Posted by: urs on May 28, 2007 3:57 PM | Permalink | Reply to this
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