n-Bundle Obstructions for Bruce

November 4, 2007

Posted by Urs Schreiber

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For every shifted central extension $1 \to \Sigma^{n-1}U(1) \stackrel{t}{\to} \hat G \to G \to 1$ of an ordinary group $G$ by the $n$ -group $\Sigma^{n-1}U(1)$ (which is trivial except in top degree, where it looks like $U(1)$ ), we can consider the problem of lifting $G$ -bundles to $\hat G$ $n$ -bundles. The obstruction to that is itself an $(n+1)$ -bundle ( $n$ -gerbe) of structure group $\Sigma^n U(1)$ .

I talked about that in

Obstructions to $n$ -bundle lifts
Obstructions, Tangent Categories and Lie $N$ -tegration
Obstructions to $n$ -bundle lifts part II, the BIG diagram
and in subsection: Obstruction Theory of String- and Chern-Simons $n$ -Transport.

The construction there is given mainly in terms of local data:

given a $G$ -1-cocycle $Y^{[2]} \stackrel{g}{\to} \Sigma G$ we may lift it to the weak cokernel $\mathrm{wcoker}(t) := (\Sigma^{n-1} U(1) \to \hat G)$ (to be thought of, I gather, as playing the role of the homotopy quotient) by postcomposing $Y^{[2]} \stackrel{g}{\to} \Sigma G \stackrel{\simeq}{\to} (\Sigma^{n-1}U(1) \to \hat G)$ and then projecting onto the cokernel of $\hat G \to (\Sigma^{n-1}U(1)\to \hat G)$ to finally obtain the obstructing $\Sigma^n U(1)$ - $n$ -cocycle $Y^{[2]} \stackrel{g}{\to} \Sigma G \stackrel{\simeq}{\to} (\Sigma^{n-1}U(1) \to \hat G) \to \Sigma^n U(1)$ (Here $Y \to X$ is a surjective submersion on base space, and all morphisms displayed are smooth pseudo anafunctors. Making this precise is straightforward either for all $n$ in the Lie $n$ -algebra version of this, or for low $n$ in the world of Lie $n$ -groupoids, and an obvious exercise (to be done) for higher $n$ in the world of Lie $n$ -groupoids).

In a discussion that began in What is the Fiber? and continued first by email and then in person at the conference in Sheffield, Bruce Bartlett complained that this construction relies entirely on local data and demanded an analogous construction on the total global objects.

Bruce pointed out that the construction of the lifting gerbe for an ordinary central extension $1 \to U(1) \to \hat G \to G \to 1$ in Brylinski’s book amounts to replacing the fibers $P_x$ of an ordinary $G$ -bundle (which are $G$ -torsors) to 2-fibers given by the groupoids of $\hat G$ -torsors over $P_x$ . These happen to be $\Sigma U(1)$ -2-torsors and hence indeed form the fibers of a $\Sigma U(1)$ -2-bundle.

I promised to think about how this fits into the big picture. Here is what I think the answer is.

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A principal $G$ -bundle (possibly with connection) on $X$ is a functor $\array{ P_1(X) \\ \downarrow^{\mathrm{tra}} \\ G\mathrm{Tor} }$ with the special property that it is smoothly locally trivializable, meaning that it can be completed to a square

$\array{ P_1(Y) &\stackrel{\pi}{\to}& P_1(X) \\ \downarrow^{\mathrm{triv}} &\Downarrow^{t}& \downarrow^{\mathrm{tra}} \\ \Sigma G &\stackrel{i}{\to}& G\mathrm{Tor} }$

such that the descent data obtained from this is smooth.

In order to decouple the discussion from what happens to the connection data, and concentrate attention here to just the $n$ -bundles themselves, I now restrict attention to the case where we take all paths to be constant. And I’ll simply write $X$ and $Y$ for the spaces regarded as discrete $n$ -categories.

For $1 \to \Sigma^{n-1}U(1) \stackrel{t}{\to} \hat G \to G \to 1$ any shifted central extension, we should be able to construct from the $G$ -bundle $\array{ X \\ \downarrow^{\mathrm{tra}} \\ G\mathrm{Tor} }$

canonically the corresponding $n$ -bundle obstruction $\array{ X \\ \downarrow^{\mathrm{tra}} \\ \Sigma^n U(1)\mathrm{Tor} } \,.$

The trick to play is now is to keep following what looks like a long exact sequence in homotopy, at the level of $n$ -groups:

$\Sigma^{n-1} U(1) \to G \hookrightarrow (K \to G) \to \Sigma^n U(1) \simeq (\hat G \to G)$

using the equivalence $\Sigma^{n}U(1) \simeq (\hat G \to G)$ (at the level of smooth $n$ -groups realized in terms of anafunctors) where $(\hat G \to G)$ is the corresponding crossed module For $n=1$ this is literally the obvious crossed module. For higher $n$ it is the analogous crossed $n$ -module, not to be discussed here. I’ll focus on $n=1$ now.

Notice that we have canonical inclusions $G \hookrightarrow (\hat G \to G)$ (the identity on objects) as well as $\Sigma^{n}U(1) \hookrightarrow (\hat G \to G)$ (identifying $\Sigma^n U(1)$ with the fiber over 1).

The construction which I am about to make amounts to accordingly embedding $G$ -1-bundles as well as $\Sigma^n U(1)$ - $n$ -bundles into the world of $(\hat G \to G)$ -n-bundles: in that world the given $G$ -1-bundle will be equivalent to some $\Sigma^n U(1)$ -n-bundle which is the obstruction to lifting it to a $\hat G$ -bundle.

What might sound intricate here is actually just the general principle behind a standard fact that people working with bundle gerbes use all the time: many people satisfy all their gerby needs entirely in terms of $PU(H)$ -bundles: these bundles uniquely represent the classes of the lifting gerbes obstructing their lift to $U(H)$ -bundles (see for instance these authors). This kind of thinking is useful for various purposes. For instance the associated $K(H)$ -bundles of algebras of compact operators may be thought of as the 2-vector bundles associated to the given gerbes: think of each algebra as a placeholder for the 2-vector space given by its category of modules.

Anyway, here is the construction in detail.

It amounts to noticing that there is a canonical injection

$\array{ G\mathrm{Tor} \\ \downarrow \\ (\hat G \to G)\mathrm{Tor} }$

which formalizes the construction that Bruce distilled out of the discussion in Brylinski’s book:

each $G$ -torsor $T$ (my $G$ -torsors here are “torsors over a point”(!) since I am following John Baez instead of most of the rest of the world in that respect) is sent to the $(\hat G \to G)$ -2-torsor that is given by the groupoid of of $\hat G$ -torsors over $G$ .

So an object in that $(\hat G \to G)$ -torsor is a $\hat G$ -torsor $\hat T$ together with its projection down to a $G$ -torsor $\array{ \hat T \\ &\searrow \\ && T }$ and a morphism is a $\hat G$ torsor morphism $\hat T \stackrel{f}{\to} \hat T'$ respecting that projection, such that we get a commuting triangle $\array{ \hat T &\stackrel{f}{\to}& \hat T' \\ &\searrow \swarrow & \\ & T } \,.$

One checks that this groupoid has a canonical $(\hat G \to G)$ action which makes it a $(\hat G \to G)$ -2-torsor.

Now, moreover this injection fits into this square

$\array{ \Sigma G &\hookrightarrow& G\mathrm{Tor} \\ \downarrow &\Downarrow^{\simeq}& \downarrow \\ \Sigma (\hat G \to G) &\hookrightarrow& (\hat G \to G)\mathrm{Tor} } \,,$

where the transformation filling it is non-canonically given, essentially, by choosing for each $\hat G$ -torsor a trivialization. Since there is no smoothness condition or anything involved at this point, this thing exists (otherwise, if we want this to be smooth in some sense or other, we can apply the standard machinery like using anafunctors et al – but the point of the $n$ -transport formalism which I am using here being that I don’t need to do that here, if I don’t want to, since I don’t need a smooth structure on the global codomain of my $n$ -transport, just on its descent data).

So, we can now send any $G$ -bundle

$\array{ X \\ \downarrow^{\mathrm{tra}} \\ G\mathrm{Tor} }$

canonically to the corresponding $(\hat G \to G) \simeq \Sigma^n U(1)$ - $n$ -bundle

$\array{ X \\ \downarrow^{\mathrm{tra}} \\ G\mathrm{Tor} \\ \downarrow \\ (\hat G \to G)\mathrm{Tor} } \,.$

For that simple construction to be admissable in the world of $n$ -transport, we need to check that this composite still has smooth local $(\hat G \to G)$ -trivializations. But it does, we simply paste any smooth local trivialization of the given $G$ -bundle

$\array{ Y &\stackrel{\pi}{\to}& X \\ \downarrow^{\mathrm{triv}} &\Downarrow^{t}& \downarrow^{\mathrm{tra}} \\ \Sigma G &\stackrel{i}{\to}& G\mathrm{Tor} }$

with the square which we just discussed, to obtain

$\array{ Y &\stackrel{\pi}{\to}& X \\ \downarrow^{\mathrm{triv}} &\Downarrow^{t}& \downarrow^{\mathrm{tra}} \\ \Sigma G &\stackrel{i}{\to}& G\mathrm{Tor} \\ \downarrow &\Downarrow^{\simeq}& \downarrow \\ \Sigma (\hat G \to G) &\hookrightarrow& (\hat G \to G)\mathrm{Tor} } \,,$

which is indeed a smooth local trivialization of our $(\hat G \to G)$ - $n$ -bundle.

This is the global construction of the obstructing $n$ -bundle that Bruce was asking for in the world of $n$ -transport.

But it might be noteworthy that this abstract nonsense reduces in terms of local data to the following simple statement:

we check that the $n$ -cocycle of the $(\hat G \to G)$ - $n$ -bundle obtained as above is simply nothing but the original $G$ -1-cocycle of the $G$ -bundle we started with, but now regarded as a $(\hat G \to G)$ - $n$ -cocycle under the canonical inclusion $G \hookrightarrow (\hat G \to G) \,.$

Hence the “real work” happens not before we take the $(\hat G \to G)$ -bundle constructed as above and realize it concretely as an equivalent $\Sigma^n U(1)$ - $n$ -bundle, using the other injection $\Sigma^n U(1) \hookrightarrow (\hat G \to G)$ already mentioned before.

Doing this computation amounts to finding a pseudonatural transformation tin can diagram (a prism for $n=1$ ) connecting the $G$ -cocycle triangles $\array{ && \bullet \\ & \multiscripts{^{g_{ij}}}{\nearrow}{} &\Downarrow^=& \searrow^{g_{jk}} \\ \bullet &&\stackrel{g_{ik}}{\to}&& \bullet }$

with the $\Sigma^n U(1)$ -triangles (thought and drawn for $n=1$ now) $\array{ && \bullet \\ & \multiscripts{^{\mathrm{Id}}}{\nearrow}{} &\Downarrow^{f_{ijk}}& \searrow^{\mathrm{Id}} \\ \bullet &&\stackrel{\mathrm{Id}}{\to}&& \bullet } \,.$

But finding and choosing these prisms is precisely the operation encoded in the composition

$Y^{[2]} \stackrel{g}{\to} \Sigma G \stackrel{\simeq}{\to} (\Sigma^{n-1}U(1) \to \hat G) \to \Sigma^n U(1)$

which expresses, as mentioned at the beginning, the obstruction directly as an operation on local data.

Posted at November 4, 2007 4:33 PM UTC

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14 Comments & 1 Trackback

Re: n-Bundle Obstructions for Bruce

Do I understand correctly that you assume that the ordinary group G does have some central extension by U(1)? So for groups which do not possess central extensions, like the diffeomorphism group in d > 1 dimensions and most of its subgroups, the discussion in this post does not apply?

Posted by: Thomas Larsson on November 5, 2007 6:12 AM | Permalink | Reply to this

Re: n-Bundle Obstructions for Bruce

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Do I understand correctly that you assume that the ordinary group $G$ does have some central extension by $U(1)$ ?

Nonabelian extensions can in principle be addressed in an analogous manner, but the discussion becomes much more intricate.

The reason is that the above makes crucial use of the fact that the cokernel of the injection $G \to (U(1) \to G)$ of 2-groups is strictly the 2-group $(U(1) \to 1) \,.$ If we replace $U(1)$ here by some non-abelian $H$ , the naive $(H \to 1)$ fails to be a 2-group.

I am expecting that the cokernel still exists in some sense, but potentially as an $n$ -group for high (possibly infinite) $n$ .

The reason for this expectation comes from looking at the corresponding Lie $n$ -algebras, which are easier to handle for high $n$ :

The inclusion $g \hookrightarrow (g \stackrel{\mathrm{Id}}{\to} g)$ of a Lie algebra $g$ into the weak cokernel of the identity map on it plays the role of the inclusion $G \hookrightarrow E G$ of the fiber into the universal principal $G$ -bundle. But with the 2-group structuire of $E G$ taken care of. So it is rather like the inclusion $\Sigma G \hookrightarrow \Sigma E G \,.$ This is where it becomes problematic: while we can form $G \to E G \to B G$ the classifying space $B G$ fails to have a group structure unless $G$ is abelian. As a result, the naive cokernel $g \to (g \to g) \to \Sigma g$ fails to make sense. But we know what does play its role: we can take the cohomology $H^\bullet(B G)$ and regard this graded ring as an abelian $n$ -group for $n$ the degree of the highest degree generator of $H^\bullet(B G)$ . For compact $G$ this is the highest degree indecomposable invariant polynomial on $g = \mathrm{Lie}(G)$ .

I started calling the corresponding Lie $n$ -algebra $b g$ . Then we can at least form $g \to (g \to g) \to b g \,.$

This is maybe more familiar in the dual qDGCA incarnation, where it reads $Chevalley-Eilenberg \leftarrow Weil \leftarrow H^\bullet(B G) \,.$

I am not sure yet if and in which sense $b g$ is really the cokernel of $g \to (g \to g)$ , but it certainly is in the cokernel.

(If anyone can offer any help on this, I’d greatly appreciate it.)

So, in a word: obstruction theory for nonabelian extensions should certainly exist, but it is much harder to get under control.

Posted by: Urs Schreiber on November 5, 2007 8:58 AM | Permalink | Reply to this

Re: n-Bundle Obstructions for Bruce

An abelian extension of a group G by some space V is an exact sequence

1 –> V –> G^ –> G –> 1.

This says that V carries a G rep, i.e. V is a module, and that the cocycle condition (gg’)v = g(g’v) holds for all g, g’ in G, v in V. V could then have some extra structure, e.g. a group structure or, I suppose, an n-group structure. But that can only lead to extra obstructions. As a first step to study the general non-abelian case, we must check that we get an abelian extension by forgetting the extra structure.

My point is that such extensions have been classified for most interesting groups, in particular for groups and subgroups of diffeomorphisms (or more precisely, for algebras of polynomial vector fields), and there are only a few possibilities. In particular, many groups do not admit central extensions, i.e. extensions by the trivial module. A classification of central extensions in the super case was made by some string theorists (Schwarz and Scherk, perhaps) back in 1976. IIRC, the result is essentially that the only Lie superalgebras with central extensions are the contact superalgebras K(1|N) with N <= 4, i.e. the N-extended super-Virasoro algebras.

Posted by: Thomas Larsson on November 5, 2007 10:29 AM | Permalink | Reply to this

Re: n-Bundle Obstructions for Bruce

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Hi Urs,

Thanks for this post. I’ve been trying to understand it all morning; I’ve had to remind myself about the basic facts of crossed modules, 2-groups etc. What I’m confused about right now is the opening line : what is a shifted central extension?

I’ve looked at my notes from Bakewell when I think you explained this to me; I think I know what it is but I need to be sure.

Let me be sure I understand what an exact sequence of higher $n$ -groups is:

(1)

1 \rightarrow \Sigma^{n-1} U(1) \stackrel{t}{\rightarrow} \hat{G} \rightarrow G \rightarrow 1

Let’s set $n=2$ . Then we’re talking about a sequence of 2-groups, such that the image of the one is isomorphic to the weak kernel of the other… something like that, right? Can you remind me?

Also, let me be sure I have the bigger picture right. Is it true that a cool example of a shifted extension of a group by $U(1)$ is the String 2-group? For every integer $k \in \mathbb{Z}$ , we have the exact sequence of 2-groups

(2)

1 \rightarrow \Sigma U(1) \rightarrow String_k (G) \rightarrow G \rightarrow 1

where $G$ is just thought of as a discrete 2-group. In other words, $String_k(G)$ is a “puff-up” of $G$ in the sense that each element $g \in G$ is now “exploded” into a groupoid $\mathcal{G}_g$ , whose objects are the paths from $1$ to $g$ and whose morphisms are the discs, bla bla, the main point being that the hom-sets in $\mathcal{G}_g$ are $U(1)$ -torsors, so that $String_k(G)$ can be regarded as a “shifted” extension of $G$ . Is that right?

Posted by: Bruce Bartlett on November 5, 2007 2:06 PM | Permalink | Reply to this

Re: n-Bundle Obstructions for Bruce

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what is a shifted central extension?

I made that term up, in lack of anything better.

All I mean is (as you have understood corectly I think): an extension not necessarily by an abelian group $A$ , but by the $(n-1)$ -fold shifted version $\Sigma^{n-1}A$ of an abelian group. This is supposed to denote the (strict) $n$ -group which is trivial except for its $(n-1)$ -morphisms, which form the space underlying $A$ , have their composite in all possible directions be given by the product in $A$ and have the strict $n$ -group product being simply the product in $A$ .

Then, by an “extension” I mean a short exact sequence of $n$ -groups, where in the examples I talked about the rightmost group, that being extended, is just an ordinary group, regarded as an $n$ -group in a trivial way.

If you force me to make precise what I mean by a short exact sequence of $n$ -groups, I’ll say that much is known about this up to $n=2$ (see for instance Bourne and Vitale, kindly pointed out to me by Mathieu Dupont here) and that the case I care most about is the String extension, which is even an extension of strict 2-groups as we explain in from loop groups to 2-groups, and that it should be straightforward but potentially tedious to spell out the obvious conditions for higher $n$ , and that I think we should be able to go up to $(n=3)$ this way for practical purposes, and that I would instead prefer to do the higher $n$ version at the level of Lie $n$ -algebras.

Then we’re talking about a sequence of 2-groups, such that the image of the one is isomorphic to the weak kernel of the other… something like that, right? Can you remind me?

Yes. See Bourne and Vitale for the general kind of definition, and notice that for the String-extensions, which we can concentrate on as long as we have $n=2$ and $G$ being compact simple and simply connected, we can even assume everything to be strictly exact.

Is it true that a cool example of a shifted extension of a group by U(1) is the String 2-group?

That’s right! And if we assume $G$ to be simple, compact and simply connected, this is already the only example.

But there should be other cool examples waiting to be discussed here, like shifted central extensions of the $n$ -torus $T^n$ . As Ulrich Bunke explained in Sheffield there is a way to look at topological T-duality slickly as an operation on a $\hat G$ -2-bundle, where $\hat G$ is a shifted central extension $1 \to \Sigma U(1) \to \hat G \to T^n \to 1$ of the $n$ -torus group ( $\simeq U(1)^n$ ).

As for the strict version of the String 2-group, this should come from the multiplicative bundle gerbe on $T^n$ . And topological T-duality is then nothing but categorified Pontryagin duality applied to such a 2-bundle.

That ought to be pretty cool, once spelled out completely. I guess I am supposed to get into contact with Ansgar Schneider about this, but haven’t yet.

$\mathrm{String}_k(G)$ is a “puff-up” of $G$ […]

That’s right! The strict version of the String Lie 2-group is first of all a huge puff-up of $G$ , and then on top of that a little central extension.

Conversely, it is a great way to “puff-down” Kac-Moody central extensions: whenever you see one in nature (like when computing the index of the Dirac operator twisted by a Vertex operator algebra bundle to obtain the Witten genus) chances are that you are secretly seeing the String puff-up, which is really just a group $G$ and a cocycle on it.

Posted by: Urs Schreiber on November 5, 2007 7:57 PM | Permalink | Reply to this

Re: n-Bundle Obstructions for Bruce

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what is a shifted central extension?

Is it fair to say that a shifted central extension of $G$ , in the sense we’ve been describing,

(1)

1 \rightarrow \Sigma^{n-1} A \rightarrow \hat{G} \rightarrow G \rightarrow 1

is exactly a geometric description of an $(n+1)$ -cocycle in $Z^2(G, A)$ ? I’m sure this is probably in John’s lecture notes on n-categories and cohomology somewhere, but I couldn’t find it.

In other words, geometrically speaking, a cocycle in $Z^{n+1}(G, A)$ is a way to take an ordinary group $G$ and an abelian group $A$ , and make an $n$ -group $\hat{G}$ which has $G$ in the 0-cells and $A$ in the $(n-1)$ -cells?

Is that right? How does it correspond with the classical Eilenberg-Maclane description of $n$ -cocycles?

Posted by: Bruce Bartlett on November 7, 2007 12:47 AM | Permalink | Reply to this

Re: n-Bundle Obstructions for Bruce

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Ok I think I’m getting a bit into the swing of things now. I have two comments to make, which I’ll make in seperate posts.

Hence the “real work” happens not before we take the $(\hat{G} \rightarrow G)$ -bundle constructed as above and realize it concretely as an equivalent $\Sigma^n U(1)$ -n-bundle, using the other injection

Mmm. Let’s consider $n=2$ . We are given a Lie group $G$ , and a principal $G$ -bundle $P \rightarrow X$ over a space $X$ . The exam question is something like : “Construct the 2-gerbe living over $X$ which measures the obstruction to lifting $P$ to a $String_k (G)$ 2-bundle”.

By a 2-gerbe on $X$ I’m thinking of a gadget which smoothly assigns to each point $x \in X$ a 2-groupoid $\mathcal{G}_x$ ,

(1)

x \mapsto \mathcal{G}_x

with the proviso that the 2-hom-sets in $\mathcal{G}_x$ are $U(1)$ -torsors. That might not be a good way of thinking about 2-gerbes… but it’s the framework in which the question is set.

So the sort of answer I would regard as “an answer” is an explicit formula which tells me what those 2-groupoids $\mathcal{G}_x$ are, for $x \in X$ . I think you’ve just about got it up in your post, but I would still like an explicit formula (I like to work from the bigger picture backwards ). At the moment we are saying something like:

“ $\mathcal{G}_x$ is the 2-groupoid whose objects are the such and such groupoids sitting over $P_x$ , whose morphisms are the such-and-such functors between them, and whose 2-morphisms are the such-and-such natural transformations”.

Can you fill in these blanks?

Posted by: Bruce Bartlett on November 5, 2007 3:05 PM | Permalink | Reply to this

Re: n-Bundle Obstructions for Bruce

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Can you fill in these blanks?

Right, that’s the point I did not spell out in the above entry, where I concentrated on the $n=1$ -example.

I think that we need to do precisely the op-version of the way we compute the weak cokernels (now here we are computing weak kernels).

Unfortunately, right now our institute’s server is down, which means that I cannot point you to the picture that I was using here.

But if you recall what that pictutre was like, or if you wait until it re-appears, then

- op that picture by reversing all arrows

- replace the bullet at the tip by a $G$ 2-torsor

- replace the other two bullets by $\hat G$ -2-torsors

and then see the obvious crank appear and turn it.

I’ll try to spell that out in more detail. Maybe tomorrow afternoon.

Posted by: Urs Schreiber on November 5, 2007 8:14 PM | Permalink | Reply to this

Re: n-Bundle Obstructions for Bruce

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…and then see the obvious crank appear and turn it.

Have you seen the wikipedia entry for crossed modules? They give a nice example about the connection between crossed modules and homotopy rel boundary.

…whenever you see one in nature (like when computing the index of the Dirac operator twisted by a Vertex operator algebra bundle to obtain the Witten genus

There is a cool Index Theory seminar running at Sheffield. I’ve found out about John Roe’s book which is written in just the kind of way I might possibly have a chance of understanding.

It’s kind of weird : have you noticed that the formula for the Dirac operator is just like the “2-character” or the familiar “trace” string diagram?

(1)

D : C^\infty(S) \stackrel{\nabla}{\rightarrow} \stackrel{metric}{\rightarrow} \stackrel{\multiply}{\rightarrow} C^\infty(S)

Copairing, do something, multiply.

Posted by: Bruce Bartlett on November 5, 2007 10:21 PM | Permalink | Reply to this

Re: n-Bundle Obstructions for Bruce

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have you noticed that the formula for the Dirac operator is just like the “2-character”

No I haven’t!! And I still don’t. But this sounds mighty interesting.

Copairing, do something, multiply.

I don’t seem to see what you seem to be seeing here. But I’ll keep that in mind and see what my subconciousness does with it…

But maybe you could give more details?

Understaning what Dirac operators really are is the single most important thing on my list of “things to be understood eventually”.

I have some ideas, and the word “pairing” is certainly involved. But right this moment I cannot quite connect this to 2-characters.

Posted by: Urs Schreiber on November 7, 2007 7:14 PM | Permalink | Reply to this

Re: n-Bundle Obstructions for Bruce

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Hi Bruce,

this morning Andrew Stacey gave me a quick intro into his work on Dirac operators on loop space. Strikingly, he told me the best way to think of the symbol map (that thing sending sections of cotangent bundle times spin bundle to spin bundle) as a generalized trace.

I immediately thought of the above message of yours. But when I mentioned to Andrew that you said this map is like a 2-character, he said he wasn’t familiar with that way of looking at it.

But I am really struck. What’s going on here?

I need to think about this on the plane back to Hamburg. We had been in a terrible rush this morning, since I had to leave to catch our bus to the train. (Now I have a few seconds in the hotel before the bus arrives :-).

I need to look at the notes that I took very quickly while Andrew was talking. But if you know at all what he is referring to, concerning how the symbol map is really to be thought of as a genralized trace, maybe you can think about how that would relate to your idea about 2-characters. Maybe there is something deep going on here. That would be cool.

More later, when there is more time (if that should ever happen again…)

Posted by: Urs Schreiber on November 9, 2007 9:44 AM | Permalink | Reply to this

“Lie-algebra only” version of String 2-group?

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I’d be interested to see if one could write down a “Lie-algebra only” version of the String 2-group. Has this been done? I’m sure there’d be a nice “dg”-style way to write it down (See below).

Suppose $G$ is a Lie group and $g$ its Lie algebra. Recall the fundamental truth : although there is no good correspondence between elements of $G$ and elements of $g$ , there is a natural one-to-one correspondence between the set of paths $P_G$ in the group (starting at the identity) and the set of paths $P_g$ in the Lie algebra (starting and ending arbitrarily):

(1)

differentiate : P_G \leftrightarrow P_g : integrate.

I spoke about this in a previous post. So the moral is : anything that can be phrased in terms of paths in the Lie group can equivalently be phrased in terms of paths in the Lie algebra.

What does the String 2-group look like in this language? Remember how it works : an object of $String_k(G)$ is a path $\gamma : 1 \rightarrow a$ in the group from $1$ to some element $a \in G$ (i.e. an object is “a group element, together with how you got there ”). A morphism $\theta : \gamma_1 \rightarrow \gamma_2$ is an equivalence class of pairs $(H, \alpha)$ consisting of a smooth homotopy of paths $H$ (fixing the endpoints) from $\gamma_1$ to $\gamma_2$ , and $\alpha \in U(1)$ measures the “twist” given by integrating the fundamental 3-form over the square, etc, etc.

What does that look like when written out exclusively in terms of paths in the Lie algebra? In other words, what is the “look mom, no group!” version?

Remember how it works : if you want to know whether two paths (starting and ending arbitarily) $\xi_1$ and $\xi_2$ in the Lie algebra will integrate to give the same Lie group element, you have the following options:

1. Duistermaat-Kolk style (see their book) : $\xi_1$ and $\xi_2$ give the same group element if and only if their difference averages to zero,

(2)

\int (\xi_2 - \xi_1 (t)) dt = 0.

This is trickier than it looks, I actually don’t understand this… can someone explain it to me?

2. Segal style (see his lecture notes): $\xi_1$ and $\xi_2$ give the same group element if and only if they can be joined by a smooth family of paths $\{\xi_s\}$ such that the Maurer-Cartan equation is satisfied for some

(3)

\eta : [0,1] \times [0,1] \rightarrow Lie(G).

To be honest, I don’t understand this one either ! I suspect its secretly the same condition as Dusitermaat and Kolk… can someone explain it?

I like the Segal version, because I like the Maurer-Cartan equation… it gets you into the $dg$ -world, which is a good place to be if you’re talking about Lie n-algebras anyway.

To summarize : I’d like to see a “look mom, no group” version of the String 2-group such that :

(a) it is phrased entirely in terms of paths in the Lie algebra $g$ , (b) the words “Maurer-Cartan equation” enter in the description.

Posted by: Bruce Bartlett on November 7, 2007 12:39 AM | Permalink | Reply to this

Re: “Lie-algebra only” version of String 2-group?

Sorry, I gave the wrong Google Books link to Segal’s lecture notes. Here is the right one : Lectures on Lie Groups and Lie Algebras, Roger Carter, Graeme Segal, Ian Macdonald.

Posted by: Bruce Bartlett on November 7, 2007 12:55 AM | Permalink | Reply to this

Re: “Lie-algebra only” version of String 2-group?

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Hi Bruce,

thanks for all the very interesting comments!

Since I am travelling at the moment, I cannot do your comments justice right now.

But here is a quick reply to

I’d be interested to see if one could write down a “Lie-algebra only” version of the String 2-group.

Unless I am misunderstanding what you are asking for, the answer is YES. Ezra Getzler and André Henriques have discussed this. See the references in On Lie $N$ -tegration and Rational Homotopy Theory.

Also, I think we talked about pretty much this remark of yours before, and I think all the comments I made then still apply.

But let me know if you think that I am missing your point.

Posted by: Urs Schreiber on November 7, 2007 7:05 PM | Permalink | Reply to this

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