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October 2, 2007

Cohomology of the String Lie 2-Algebra

Posted by Urs Schreiber

Unfortunately I haven’t found the time to come by the nn-Café a lot lately. After I returned from my travels I needed to recover a little and see my family. Then, to my considerable delight, Danny Stevenson arrived last weekend in Hamburg, where he now has a position in our department. We spent the better part of the last two days taking care of the inevitable administrative paperwork and with running around in Hamburg trying to find a nice place for him to stay.

While sitting on trains through and in Cafés in Hamburg, we had lots of time for discussion. In one of these discussions the following insight materialized, which I believe I am allowed to share. It’s rather beautiful in its simplicity, and indeed won’t be news at all to experts – except possibly for the slightly new point of view which it might offer on a well-known construction.

I will descibe how the Lie 2-algebra cohomology of the String Lie 2-algebra of Baez-Crans Lie nn-algebra type is governed by the twisted 2\mathbb{Z}_2-graded differential

d H:=d+H,, d_H := d + H \wedge \cdot ,,

familiar from the study of twisted K-theory and obtainable for any closed 3-form HH, for the case where we are looking at differential forms on the underlying compact Lie group GG with HH being the canonical 3-class on that group.

It’s mostly – but not entirely – a big tautology. But possibly an enjoyable and insightful one.

All you need to know is that, following the general reasoning outlined in String- and Chern-Simons nn-Transport, the String Lie 2-group corresponding to some semisimple group GG is something which admits a graded differential algebra of something like left-invariant 1-forms, that looks as follows:

as a graded commutative algebra it is

(sg *ss *), \wedge^\bullet ( s g^* \oplus s s \mathbb{R}^* ) \,,

which is supposed to mean that it is the exterior algebra generated freely from ordinary left-invariant differential forms on GG together with one single generator in degree 2 (which you might think of either as an auxiliary 2-form, or, better, as a 1-form on a space of morphisms). This extra generator I’ll denote by

b. b \,.

A typical homogeneous element in this algebra hence look like

ωbb, \omega \wedge b \wedge b \,,

where ω\omega is any left-invariant pp-form on GG.

There is a +1-graded differential dd on this algebra, which squares to zero, d 2=0d^2 = 0. It is defined to act on elements ω\omega entirely built from left-invariant 1-forms on GG in the standard way. On the single generator bb, however, it is defined to act as

db:=μ=H. d b := \mu = H \,.

Here the right hand side denotes the canonical 3-cocycle

μ=,[,] \mu = \langle \cdot, [\cdot,\cdot] \rangle

on the semisiple Lie algebra. In order to emphasize that I want to think of this as extended to a left-invariant 3-form on GG I’ll equivalently write this as HH.

Some readers will find this extremely suggestive. If you are not among those, just ignore this and accept the fact that I call the differential of bb by the name HH, and that I assume that HH is a closed left invariant 3-form on GG.

The differential graded commutative algebra which we have obtained this way is the Koszul-dual to the String Lie 2-algebra. More on the general background underlying these considerations can be found in

String- and Chern-Simons nn-Transport

The general top-degree left invariant form on the String 2-group hence looks like

ω=ω n+ω n2b+12ω n4bb+16ω n6bbb+, \omega = \omega_n + \omega_{n-2} \wedge b + \frac{1}{2}\omega_{n-4} \wedge b \wedge b + \frac{1}{6}\omega_{n-6} \wedge b \wedge b \wedge b + \cdots \,,

where ω n\omega_{n} is supposed to denote a left invariant nn-form on GG, for nn the dimension of GG, and similarly for all the other ω k\omega_k that appear.

Hence by expanding a general top degree form on the String 2-group like this, it becomes a mere array of ordinary forms on the ordinary group, one for each even or one for each odd degree. We may similarly look at top-minus-one degree forms this way. But then nothing further is gained by looking at top-minus-two degree and so on.

Hence we find that these forms on the Lie 2-group may be thought of as coming from inhomogeneous but ordinary left invariant forms sitting in a 2\mathbb{Z}_2-grading.

Next we can – easily – work out the differential on the Lie 2-group acting on our forms as above. By using the only two rules we have, which say that

- dd acts on the ω k\omega_k as usual

- dd acts on the generator bb by sending it to the 3-form HH

we immediately find

dω=(dω n2+Hω n4)+12(ω n4+Hω n6)b+, d \omega = (d \omega_{n-2} + H \wedge \omega_{n-4}) + \frac{1}{2}(\omega_{n-4} + H \wedge \omega_{n-6}) \wedge b + \cdots \,,

But this means nothing but that the differential on left-invariant forms on the Lie 2-group acts like the twisted differential

d+H d + H \wedge \cdot

on 2\mathbb{Z}_2-graded inhomogeneous ordinary left invariant forms on the ordinary group.

Accordingly, the two cohomologies coincide. And the properties of the operator d+Hd + H \wedge \cdot have attracted quite some attention.

Posted at October 2, 2007 9:45 PM UTC

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8 Comments & 1 Trackback

Re: Cohomology of the String Lie 2-Algebra

Hi Urs, good to see you back. I don’t quite understand what’s going on here, but I can see it’s interesting stuff and I am hoping you can explain it to me when we meet up at the upcoming confrence.

But I’ll abuse this posting opportunity to ask a general question about Lie groups and Lie algebras!

I’ve just attended a lecture by Kirill Mackenzie on the integration of Lie algebras la Duistermaat and Kolk’s book . This method is called “integration by paths” and I find it very elegant.

I’d like to ask geometers at the cafe : how well-known is it?

This is how it works. We have a Lie algebra gg (how does one get the mathfrak symbol?), and we’re going to construct the simply connected Lie group GG whose Lie algebra is gg.

Let P(g)P(g) be the space of smooth paths in the Lie algebra, starting and ending arbitrarily. Now, given some path in a vector space, we can always integrate it to give a path in the endomorphisms of that vector space starting at the identity. For ξP(g)\xi \in P(g), we write the resulting path-in-the-endomorphisms as

(1)A ξ:[0,1]End(g). A_\xi : [0,1] \rightarrow End(g).

Now, the key point is to observe that in the world of connected Lie groups and Lie algebras, although there isn’t a nice correspondence between points of gg and points of GG (the exponential map isn’t injective or surjective in general), there is a canonical bijection between paths in gg and paths in GG, in the obvious way.

And indeed, pointwise multiplication of paths in GG corresponds to the following multiplcation of paths ξ 1,ξ 2P(g)\xi_1, \xi_2 \in P(g):

(2)ξ 1ξ 2=ξ 1+A ξ 1(ξ 2). \xi_1 \cdot \xi_2 = \xi_1 + A_{\xi_1} (\xi_2).

So P(g)P(g) is a group. Now we have all the tools we need!

Integration. To integrate gg into a Lie group GG, we simply set

(3)G:=P(g)/ G := P(g) / \sim

where we define two paths ξ 1,ξ 2P(g)\xi_1, \xi_2 \in P(g) to be equivalent if their difference “averages to zero”, i.e.

(4) 0 1(ξ 2(t)ξ 1(t))dt=0 \int_0^1 (\xi_2(t) - \xi_1(t)) dt = 0

Here we are simply doing a standard vector calculus integral.

That’s it! That’s how easy it is to integrate a Lie algebra! I think it’s pretty cool!

How well-known is this? I’ve seen a related integration-by-paths procedure by Graeme Segal in the book Lectures on Lie Groups and Lie Algebras except there he uses the “concatenate and translate” multiplication of paths instead of pointwise multiplication. I don’t know how to relate the two.

Posted by: Bruce Bartlett on October 4, 2007 4:05 PM | Permalink | Reply to this

Re: Cohomology of the String Lie 2-Algebra

Hi Urs, good to see you back.

Hi Bruce, thanks a lot for your message.

I am still not quite back yet, but it should get better soon. Am looking forward to getting back to discussion here.

This method is called “integration by paths”

Interesting. Sounds vaguely similar to the method of integrating Lie nn-algebroids by mapping tangent algebroids of kk-disks into them that we have been discussing a bit recently (last time here). But then, it also sounds quite different. I am not sure yet.

Currently I feel that the truly good way to integrate Lie nn-algebras g (n)g_{(n)} is to consider the generalized smooth space

BG (n) B G_{(n)}

defined to be the sheaf

UHom nLieAlg(TU,g (n)) U \mapsto \mathrm{Hom}_{n\mathrm{LieAlg}}(T U , g_{(n)})

and then form its fundamental nn-groupoid

G (n):=Π n(BG (n)). G_{(n)} := \Pi_n( B G_{(n)} ) \,.

This is not entirely clear yet, but at least morally this seems to be what underlies Pavol Ševera’s approach, which is closely related to Henriques’ approach.

Since what you describe also involves some kind of homotopy classes of paths (though apparently in a different sense) I am hoping that it is somehow a different way to look at the same kind of phenomenon.

Now, given some path in a vector space, we can always integrate it to give a path in the endomorphisms of that vector space starting at the identity.

Could you give more details here? It seems I can think of at least two things you might mean here. Please help.

Posted by: Urs Schreiber on October 5, 2007 11:00 AM | Permalink | Reply to this

Re: Cohomology of the String Lie 2-Algebra

Could you give more details here? It seems I can think of at least two things you might mean here. Please help.

Sorry, I was a bit confused before (which I tried to disguise). I think I understand it better now.

Given a path in the Lie algebra ξP(g)\xi \in P(g), the differential equation

(1)dAdt=ad(ξ(t))A(t),A(0)=id \frac{dA}{dt} = ad(\xi(t)) \circ A(t), \quad A(0) = id

has a unique solution A ξ:[0,1]End(g)A_\xi : [0,1] \rightarrow End (g). To be completely clear, the right hand side above is the composition of two operators in End(g)End(g) - firstly A(t)A(t) and then ad(ξ(t))[ξ(t),]\ad(\xi(t)) \equiv [\xi(t), \cdot]. That is, you “start at the identity and then curve along according to the bracket of ξ\xi”.

The group operation on paths in the Lie algebra (the shadow of pointwise multiplication of paths in the Lie group) is

(2)ξ 1ξ 2=ξ 1+A ξ 1(ξ 2). \xi_1 \cdot \xi_2 = \xi_1 + A_{\xi_1} (\xi_2).

And the Lie group GG can be recovered as

(3)G:=P(g)/ G := P(g) / \sim

where we define two paths in the Lie algebra to be equivalent if their difference “averages to zero” in the standard vector calculus way, as in my first post.

I’m impressed by this procedure, it’s so simple and elegant.

Posted by: Bruce Bartlett on October 5, 2007 11:26 AM | Permalink | Reply to this

Re: Cohomology of the String Lie 2-Algebra

Interesting. Sounds vaguely similar to the method of integrating Lie n-algebroids by mapping tangent algebroids of k-disks into them that we have been discussing a bit recently.

Well, from the abstract from the talk:

There are several distinct proofs of the integrability of (finite dimensional, real) Lie algebras, but the main purpose of this seminar is to describe the proof of Duistermaat using path spaces.
This method is the foundation of the solution of the integrability problem for Lie algebroids and in retrospect can also be seen as underlying the construction by Cattaneo and Felder of a symplectic realization for any Poisson manifold, using Poisson sigma models.

So… yes there does seem to be a strong connection!

Posted by: Bruce Bartlett on October 5, 2007 11:35 AM | Permalink | Reply to this

Re: Cohomology of the String Lie 2-Algebra

Bruce wrote:

Given a path in the Lie algebra ξP(g)\xi \in P(g), the differential equation

dAdt=ad(ξ(t))A(t) \frac{d A}{d t} = \mathrm{ad}(\xi(t)) \circ A(t)

Ah, good. That’s the equation for parallel transport parallel to ξ\xi. Some people would write

A=Pexp( 0 1ξ(t)dt) A = P \exp(\int_0^1 \xi(t) d t)

and call this the path-ordered exponential.

As we had discussed here previously, the method of mapping tangent algebroids into the Lie algebra is so cool because it computes this path ordered exponential without ever computing it:

the thing is that any to ξ\xis (to be thought of as Lie algebra valued 1-forms) which are interpolated by a flat Lie algebra valued 1-form on the disk will – by the nonabelian Stokes theorem – lead to the same path ordered exponential. Hence it suffices to just look at the paths in the Lie algebra and not consider their exponentiation at all.

I am guessing that the equivalence condition you stated can be translated into the existence of a flat 1-form on disks this way.

Gotta run…

Posted by: Urs Schreiber on October 5, 2007 1:44 PM | Permalink | Reply to this

Re: Cohomology of the String Lie 2-Algebra

Ah, good. That’s the equation for parallel transport parallel to ξ\xi. Some people would write A=Pexp( 0 1ξ(t)dt)A = P exp(\int_0^1 \xi(t) dt) and call it the path-ordered exponential .

Right - good point. Excellent!

I am guessing that the equivalence condition you stated can be translated into the existence of a flat 1-form on disks this way.

Yes, you are probably right. You will need to explain it to me.

Posted by: Bruce Bartlett on October 5, 2007 2:24 PM | Permalink | Reply to this

Re: Cohomology of the String Lie 2-Algebra

Here is another remark on the String Lie 2-algebra cohomology:

notice how I had introduced those factorial prefactors into the expansion

ω=ω n+ω n2b+12ω n4bb+16ω n6bbb \omega = \omega_n + \omega_{n-2}\wedge b + \frac{1}{2}\omega_{n-4} \wedge b \wedge b + \frac{1}{6} \omega_{n-6} \wedge b \wedge b \wedge b

of the general degree nn String Lie 2-algebra cochain, where the omgega k\omgega_k are cohains on the ordianry underlying Lie algebra. Of course this means that we may abbreviate equivalently as

ω=( pω p)exp(b)| n \omega = (\sum_p \omega_{p}) \exp(b)|_{n}

where the exponential is with respect to the wedge product and where the | n|_n is the operation of picking the homogenoeus degree nn part out of an inhomogeneous cochain.

Written this way, the crucial relation that I pointed out

dω=((d+H) pω p)exp(b)| n d \omega = ( (d + H \wedge)\sum_p \omega_{p}) \exp(b)|_{n}

is even more manifest.

Clearly something is going on here, since expressions of the form

( pω p)exp(b)| n (\sum_p \omega_{p}) \exp(b)|_{n}

appear throughout all of D-brane physics.

It feels like we have come a tiny bit closer to understanding what this really means. But then, I definitely need to learn a couple of things here.

Posted by: Urs Schreiber on October 6, 2007 10:32 AM | Permalink | Reply to this

Re: Cohomology of the String Lie 2-Algebra

Parts of the discussion of the Lie nn-algebra cohomology of String-like Baez-Crans type Lie nn-algebras can now be found in the section

Lie nn-algebra cohomolohy

subsection

Lie nn-cohomology of g μg_\mu

in the set of slides titled: On String- and Chern-Simons n-Transport.

Posted by: Urs Schreiber on October 9, 2007 7:26 PM | Permalink | Reply to this
Read the post On Lie N-tegration and Rational Homotopy Theory
Weblog: The n-Category Café
Excerpt: On the general ideal of integrating Lie n-algebras in the context of rational homotopy theory, and about Sullivan's old article on this issue in particular.
Tracked: October 20, 2007 5:18 PM

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