Higher Gauge Theory and Elliptic Cohomology

July 31, 2007

Higher Gauge Theory and Elliptic Cohomology

Posted by John Baez

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After some fun in Greece, I’ve been holed up in Greenwich the last two days preparing my talk for the 2007 Abel Symposium. This is an annual get-together sponsored by the folks who put out the Abel prize, a belated attempt to create something like a Nobel prize for mathematicians.

One of the themes of this year’s symposium is “elliptic objects and quantum field theory”. So, while my true love is higher gauge theory, my talk will emphasize its relation to elliptic cohomology and related areas of math:

John Baez, Higher Gauge Theory and Elliptic Cohomology.

Abstract: The concept of elliptic object suggests a relation between elliptic cohomology and "higher gauge theory", a generalization of gauge theory describing the parallel transport of strings. In higher gauge theory, we categorify familiar notions from gauge theory and consider "principal 2-bundles" with a given "structure 2-group". These are a slight generalization of nonabelian gerbes. After a quick introduction to these ideas, we focus on the 2-groups $String_k(G)$ associated to any compact simple Lie group $G$ . We describe how these 2-groups are built using central extensions of the loop group $\Omega G$ and how the classifying space for $String_k(G)$ -2-bundles is related to the "string group" familiar in elliptic cohomology. If there is time, we shall also describe a vector 2-bundle canonically associated to any principal 2-bundle, and how this relates to the von Neumann algebra construction of Stolz and Teichner.

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I hope you folks — especially Urs — can look this over and find typos and other problems before I take off for Oslo on Sunday.

Urs will be pleased to see that I’ve caved in and started using his notation for the string 2-group.

He’ll also note that I’m deliberately not explaining his construction of a 2-vector bundle from a principal 2-bundle for the string 2-group — the construction that has very strong ties to Stolz and Teichner’s work on elliptic objects. I thought it would take too long at the end of a very long talk. Instead, while I provide links to his work, I’ll explained a similar construction which is simpler and ‘more canonical’ — it works easily for any 2-group. This is Alissa Crans and Danny Stevenson and I were working on this spring.

I would like to better understand the relation of the two constructions!

Posted at July 31, 2007 9:38 AM UTC

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21 Comments & 3 Trackbacks

Re: Higher Gauge Theory and Elliptic Cohomology

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Nice. Interesting construction of the 2-rep.

I’ll think about this. But not right now, I have a guest here.

One point which might be worth mentioning:

Mere 2-functors with values in $\Sigma \mathrm{String}_k(G)$ will only describe String-2-bundles which come from lifting flat $G$ -bundles.

One way to see this is that the fake-flatness condition for 2-functors $d A + A \wedge A + \partial B = 0$ with $A$ and $B$ taking values in some path algebra, translates here into a true flatness condition at the end of these paths, since $B(2\pi) = 0$ .

But it’s $A(2\pi)$ which is the original connection on the $G$ -bundle of which the $\mathrm{String}_k(G)$ -bundle is a lift.

So we do need to pass to flat $\mathrm{INN}(\mathrm{String}_k(G))$ -3-transport to get the general String-2-bundles.

But that’s not a bug, but a feature. For one $\mathrm{Lie}(\mathrm{INN}(\mathrm{String}_k(G))$ is isomorphic to the Chern-Simons Lie 3-algebra $\mathrm{cs}_k(g) \,.$ I didn’t explicitly mention it there, but one should take everything I described here and apply one more $\mathrm{inn}(\cdot)$ to it, to really get the codomains of the transport.

Posted by: Urs Schreiber on July 31, 2007 10:32 AM | Permalink | Reply to this

Re: Higher Gauge Theory and Elliptic Cohomology

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I enjoyed John’s talk very much. I am very interested and have many questions to ask, though I’ll resign myself to the following. One of the main themes I am trying to get a grip of, is making the connection between the language of higher gauge theory, 2-bundles, etc. and the language of sheaves of groupoids etc. àla Brylinski.

1. Small typos $pic$ . I think that it should be “Lie 2-group” at the Theorem on the top of page 12. Also, on page 14, last sentence, should be $|String_k (G)|$ , no? Perhaps not a typo, but I couldn’t parse the phrase on page 17 “isomorphism classes of principal $|G|$ -2-bundles”… $|G|$ is an ordinary group, what is a 2-bundle for it? Finally, top of page 18, “ $P$ the trivial principal $G$ -bundle”… don’t you mean 2-bundle?

2. After reading the “quick and dirty” description of 2-Bundles given on pages 15-16, it seems to me that a 2-bundle over $X$ is just a quick and dirty description of the descent data of a sheaf of groupoids over $X$ . I know this is probably old news to you, but I’d like to understand these things better.

A sheaf of groupoids would assign to each open set $U_i$ , a groupoid $G_i$ (I wish I could get calligraphic letters),

(1)

U_i \mapsto G_i

and to each intersection, a functor of groupoids

(2)

U_{ij} \mapsto F : G_i|_{ij} \rightarrow G_j|_{ij}

And so on.

If we want to be quick and dirty, we can choose our open covering fine enough so that the groupoids $G_i$ are all equivalent. We might as well imagine they’re all the same groupoid $G$ , and concentrate on the functors, which will now be automorphisms of $G$ . In other words, to every intersection $U_{ij}$ , we’ll be assigning an element of the automorphism 2-group $AUT(G)$ ,

(3)

U_{ij} \mapsto F \in AUT(G)

And so on. This is just the data of a principal 2-bundle.

So… recall that I want to make the connection between the language of higher gauge theory, the string 2-group $String_k(G)$ , etc. and the language of Brylinski when he defines the canonical sheaf of groupoids associated to any simply-connected simple compact Lie group.

The crucial point is thus :

If the 2-group $String_k(G)$ is to be regarded as an automorphism 2-group of some groupoid $E$ , what is the groupoid $E$ ?

Recall that Brylinski’s “E” over an open set $U \subset G$ (here $G$ is the lie group) would be the groupoid of pairs $(L, \nabla)$ where $L$ is a line bundle on $P_1 U$ and $\nabla$ is a fibrewise connection on $L$ , with curvature related to $\nu$ in a precise way (can’t quite remember).

Can you make the connection between these two pictures?

Posted by: Bruce Bartlett on August 1, 2007 3:07 PM | Permalink | Reply to this

Re: Higher Gauge Theory and Elliptic Cohomology

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Can you make the connection between these two pictures?

Like a bundle has a sheaf of sections with values in sets, a 2-bundle is supposed to have a stack of sections with values in groupoids.

The groupoid over $U$ hence plays the role of the groupoid of sections of the 2-bundle over $U$ : each object is a section, each morphism a morphism of such sections.

But, on the other hand, if we start with a non-empty and transitive stack of groupoids this way – called a gerbe (I mean, this is really a gerbe. “bundle gerbes” are not gerbes – they are just equivalent to them) – we find that it always corresponds to a 2-bundle with structure 2-group being $G_{(2)} = \mathrm{AUT}(G)$ , the automorphism 2-group of some ordinary group $G$ .

I am not sure what kind of modification of the concept of a gerbe would actually give the 2-bundles for more general 2-groups, like $\mathrm{String}_k(G)$ .

Once, a long while ago, I tried to think about this, but wasn’t really up to it then. Now I think it would be an easy question, but I’d have to start thinking about it again.

Problem is that I see little motivation for looking at 2-bundles through their gerbes of sections. Seems to be unnecessarily indirect. At least for the applications that I am concerned with.

Posted by: Urs Schreiber on August 1, 2007 5:01 PM | Permalink | Reply to this

Re: Higher Gauge Theory and Elliptic Cohomology

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Problem is that I see little motivation for looking at 2-bundles through their gerbes of sections.

Well, it would be really helpful for people like me who have been brought up (not to say that I understand it yet) on the Brylinski viewpoint of gerbey geometry, geometrical quantization, etc.

I mean, when John writes

… like when G is our favorite 2-group, the fundamental gerbe of a compact simple Lie group G.

the message seems to be that you have all these pictures sorted out, that you can elegantly translate between Brylinski-type gerbey geometry and higher gauge theory.

So I would just like to underststand explicitly the dictionary which will translate the concept of a $String_k(G)$ 2-bundle into a Brylinski-style sheaf of groupoids. What I’d really like to see is how those sheaves of groupoids - whose objects over an open set $U$ are the pairs $(L, \nabla)$ where $L$ is a line bundle on $P_1(U)$ and $\nabla$ is a connection on it whose curvature is related to the 3-form $\nu$ - enter the game, since I am rather fond of these guys.

I know that I am a bit muddled up… I’d just like to understand better the translation between higher gauge theory and Brylinski-style geometry.

Posted by: Bruce Bartlett on August 1, 2007 5:36 PM | Permalink | Reply to this

Re: Higher Gauge Theory and Elliptic Cohomology

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I mean, when John writes

…like when $G$ is our favorite 2-group, the fundamental gerbe of a compact simple Lie group $G$ .

the message seems to be that you have all these pictures sorted out,

It is of course good to understand as many pictures as possible. But notice that while John says “fundamental gerbe” here, he is being slightly imprecise: as I recalled above the groupoid $\mathrm{String}_k(G)$ is (in the true sense of “is”: canonical isomorphism instead of chosen equivalence) the level $k$ bundle gerbe on $G$ : a line bundle on the space $Y^{[2]} = P G \times_G P G$ of based baths in $G$ with certain properties, as you know.

While one certainly can translate this into the language of true gerbes (non-empty and transitive stacks in groupoids over $G$ ) I’d dare say that it will hardly make any of the constructions that John talks about more transparent.

“Line bundle gerbes” are not gerbes. There is an equivalence of a 2-category of line bundle gerbes with true $U(1)$ -gerbes, though. What “line bundle gerbes” really are is a categorified transition function - canonically isomorphic to certain 2-anafunctors.

As you can see, I think that “bundle gerbe” is a misnomer. It is like saying “ $G$ -bundle” when you mean “ $G$ -transition function” or $G$ -cocycle: both not equal but just equivalent. And not even canonically so.

Of course all this is not to say that it hurts to be able to translate between 2-bundles, “bundle gerbes” and real gerbes. But if you just want to grok what John is talking about in his lecture, I’d say that stacks in groupoids are just a distraction.

Posted by: Urs Schreiber on August 1, 2007 6:16 PM | Permalink | Reply to this

Re: Higher Gauge Theory and Elliptic Cohomology

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Thanks, Bruce, for catching all those typos! Just to confirm your sanity:

Small typos $pic$ . I think that it should be “Lie 2-group” at the Theorem on the top of page 12. Also, on page 14, last sentence, should be $|String_k (G)|$ , no?

Right both times.

Perhaps not a typo, but I couldn’t parse the phrase on page 17 “isomorphism classes of principal $|G|$ -2-bundles”… $|G|$ is an ordinary group, what is a 2-bundle for it?

You’re right, I meant to say “bundle”.

Finally, top of page 18, “ $P$ the trivial principal $G$ -bundle”… don’t you mean 2-bundle?

Yes, this time I meant to say “2-bundle”.

Thanks a lot! My talk will be much better without all these confusing slips.

After reading the “quick and dirty” description of 2-bundles given on pages 15-16, it seems to me that a 2-bundle over $X$ is just a quick and dirty description of the descent data of a sheaf of groupoids over $X$ .

Actually, what I presented was not so much the definition of a principal 2-bundle as a recipe for constructing one from ‘transition functions’ — your ‘descent data’.

But, your main point here is very much on the right track. Urs already explained the details so I’ll just say this: if you’ve got a smooth 2-group of the special form $AUT(G)$ for some smooth group $G$ , it has nonabelian $G$ -gerbe of sections.

Using this, Toby proved that the 2-category of $\AUT(G)$ -2-bundles over some space is equivalent to the 2-category of nonabelian $G$ -gerbes over that space.

So, as always in mathematics, you should not let the diversity of formalisms distract you from the shortage of truly distinct ideas!

You write:

The crucial point is thus:

If the 2-group $String_k(G)$ is to be regarded as an automorphism 2-group of some groupoid $E$ , what is the groupoid $E$ ?

Well, this may be where the roads part. I see no reason why every 2-group should be the automorphism 2-group of some groupoid, any more than every group is the automorphism group of some set! I’m pretty darn sure that $\String_k(G)$ is not the automorphism group of some group — and I don’t see how it could be the automorphism group of some groupoid, either.

This is my main (and perhaps only) beef with nonabelian $G$ -gerbes: they take the group $G$ as basic and then cook up the 2-group $AUT(G)$ from that, instead of working directly with a general 2-group. So, they miss out on some really fun examples!

Larry Breen sort of apologizes to me about this every time he hears me give a talk. He knew perfectly well that 2-groups of the form $AUT(G)$ are a special case, and he could have handled the more general case if he’d felt like it; he just didn’t think it was worthwhile.

On a slightly different thread, you write:

I mean, when John writes:

… like when G is our favorite 2-group, the fundamental gerbe of a compact simple Lie group G.

the message seems to be that you have all these pictures sorted out, that you can elegantly translate between Brylinski-type gerbey geometry and higher gauge theory.

As Urs notes, I was in a relaxed mood when I wrote that. The real message is that a couple of Ph.D. theses could sort out a bunch of this stuff. In his thesis, Toby showed that

for any smooth group $G$ , the 2-category of $\AUT(G)$ -2-bundles over some space is equivalent to the 2-category of nonabelian $G$ -gerbes over that space.

and also that

for any smooth abelian group $A$ , the 2-category of $A[1]$ -2-bundles over some space is equivalent to the 2-category of abelian $A$ -gerbes over that space.

Here $A[1]$ is the 2-group with one object, one morphism and $A$ as 2-morphisms — a ‘once shifted’ version of $A$ .

But, right now you want another Ph.D. student to write a paper detailing the relation between 2-groups and multiplicative gerbes. That would be great.

Posted by: John Baez on August 2, 2007 1:20 PM | Permalink | Reply to this

Re: Higher Gauge Theory and Elliptic Cohomology

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But, right now you want another Ph.D. student to write a paper detailing the relation between 2-groups and multiplicative gerbes. That would be great.

Konrad Waldorf, PhD student (but not for much longer) here in Hamburg is making great progress with understanding multiplicative gerbes. When he is done with what he is doing, we might better know what the analogue of the 2-group $\mathrm{String}_k(G)$ is for the cases that $G$ is not simply connected, for instance.

Posted by: Urs Schreiber on August 2, 2007 2:16 PM | Permalink | Reply to this

Re: Higher Gauge Theory and Elliptic Cohomology

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I see no reason why every 2-group should be the automorphism 2-group of some groupoid, any more than every group is the automorphism group of some set!

Yes. On the other hand, for any $G_{(2)}$ -2-bundle we get at least a pre-stack with values in groupoids of its sections. I suppose after stackified, this will turn out to be a $G$ -gerbe for some group $G$ such that its $\mathrm{AUT}(G)$ -valued transition data somehow factors through $G_{(2)}$ .

It must be something like this. Should actually be straightforward. If I had a student under my control, I would assign this task to him or her…

Posted by: Urs Schreiber on August 2, 2007 2:25 PM | Permalink | Reply to this

Re: Higher Gauge Theory and Elliptic Cohomology

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I suppose after stackified, this will turn out to be a $G$ -gerbe for some group $G$ such that…

Not so! A torsor for the gr-stack associated to a crossed module is not a gerbe. The interpretation of $H-BiTors$ as the automorphisms of $H-Tors$ is what leads to linking $H$ -gerbes (locally like $H-Tors$ ) and $H-BiTors$ -torsors. I don’t know if there is such an interpretation for a general gr-stack, but I suspect not.

Although, given the fact that a crossed module $H \to G$ has a unique map to the crossed module $H \to Aut(H)$ , a torsor such as above will have a map to an $H$ -gerbe.

Posted by: David Roberts on August 3, 2007 3:05 AM | Permalink | Reply to this

Re: Higher Gauge Theory and Elliptic Cohomology

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Not so! A torsor for the gr-stack associated to a crossed module is not a gerbe.

Yes, I agree. But consider this: given a 2-bundle for some structure 2-group, it surely has a prestack of sections, being a prestack in groupoids.

This can be stackified, and I would certainly think that the result is a transitive and locally non-empty stack in groupoids.

So, I was thinking, the stack of sections of any old crossed module 2-bundle should give rise to a gerbe. But if it is a gerbe, it must, as you point out, correspond to something involving just a group $H$ (or $G$ ) and its torsors. So therefore I guessed that we do get a gerbe for some other group, but which is somehow special in that it knows that it comes from the stack of sections of a 2-bundle for the given 2-group we started with.

I mean, the only way this can fail is that the stack of 2-sections of a 2-bundle is not locally non-empty and transitive. But that would be quite strange.

Posted by: Urs Schreiber on August 3, 2007 2:19 PM | Permalink | Reply to this

Re: Higher Gauge Theory and Elliptic Cohomology

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Here is the general set-up: given a gerbe we can always find local sections (over $U_i$ , say), and these are (after restricting further if necessary) unique up to isomorphism. This means there is a sheaf of groups $G_i$ on $U_i$ such that the gerbe is equivalent to $G_i-Tors$ over $U_i$ . If there is a sheaf of groups $G$ on our space such that $G\big|_{U_i} \simeq G_i$ , then we call our gerbe a $G$ -gerbe. But in general we are not assured that this sheaf $G$ exists.

What is necessary for this sheaf of groups to exist is that there are coherent isomorphisms $G_i\big|_{U_j} \simeq G_j\big|_{U_i}.$ These will be sections of the sheaf $\mathrm{Isom}(G_i,G_j)$ over $U_{ij}$ .

So what does this mean for 2-bundles? Let $E$ be a 2-bundle for the 2-group $G_2$ associated to the crossed module $H \to G$ . Then we have a groupoid $E$ over our space $X$ , and let us say $E$ is trivialised by a cover $U$ if there is an equivalence $E\times_X U \simeq G_2 \times U$ as a functor over $X$ . Then the stack of sections will be all possible local trivialisations (and transformations between them).

If we take the trivial 2-bundle, what to its sections look like? These will be the automorphisms of the fibre $G_2$ . This isn’t so helpful.

Another way of thinking of sections might be as generalised maps/anafunctors $X \to E$ such that the usual condition holds. In that case, a section is a cover $U \to X$ and a functor $U\times_X U \to E$ such that the triangle built from these commutes, or possibly only up to a 2-arrow.

For the trivial 2-bundle as above, this would mean that a section is an $(H \to G)-$ torsor, and so the stack of such sections is the stack of these torsors. This stack is locally non-empty, but not locally connected. I think something may have gone wrong here, because if I set $H \to G = H \to Aut(H)$ , I don’t get $H$ -gerbes!

I think we need to get a grip on what the stack of sections of a 2-bundle is before we can say what it is. My feeling is that it will be locally non-empty (as in the previous paragraph), but possibly not locally connected.

One thing we need to figure is: what is an automorphism of a section? This will give us the best idea of what is going on, regardless of my spouting in the last few paragraphs.

Posted by: David Roberts on August 6, 2007 8:35 AM | Permalink | Reply to this

Re: Higher Gauge Theory and Elliptic Cohomology

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but possibly not locally connected.

Ah, you are right! Yes, that sounds right. Of course. Okay, good. So probably general 2-bundles yield a generalization of the notion of gerbe where we relax the condition of local connectedness.

I think we need to get a grip on what the stack of sections of a 2-bundle is before we can say what it is.

As you have probably seen, I have been thinking a lot about this. I don’t think I have the final form of the answer quite figured out entirely, though.

It’s easiest for $n$ -vector $n$ -bundles: look at them in terms of their fiber assigning functor $f$ with values in $n\mathrm{Vect}$ . Let $I$ be the tensor unit in the $n$ -category of such functors. Then the space of sections of $f$ is $\Gamma(f) = \mathrm{Hom}(I,f) \,,$ i.e. just the generalized elements. This works very nicely for $n=1$ and $n=2$ . I talk about that for instance in Quantum 2-States: Sections of 2-vector bundles.

But this is apparently just a special case of a more general principle. It seems that for general principal $n$ -bundles we want to be looking at morphisms not into $f$ , but into $d f$ , the way I describe in the discussion of tangent categories, towards the end.

This might actually be related to what John keeps hinting at in his Tale of Groupoidification: passing from $\Sigma G_{(n)}$ to $T \Sigma G_{(n)} = \mathrm{INN}(G_{(n)})$ is apparently not unrelated to passing to an $n$ -vector space representation of $G_{(n)}$ .

Let’s keep thinking about this. Something important is going on here which hasn’t quite seen the light of day yet.

Posted by: Urs Schreiber on August 6, 2007 10:27 AM | Permalink | Reply to this

Re: Higher Gauge Theory and Elliptic Cohomology

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As usual, on the train home last night I remembered more I should have written ;)

The stack of sections of $X \times G_2$ is of course the automorphisms of a trivial 2-bundle. When $G_2 = AUT(G)$ we know this as $G-Bitors = Aut(G-Tors)$ , so the trivial 2-bundle in that case of $G-Tors$ , the trivial gerbe.

Posted by: David Roberts on August 7, 2007 7:27 AM | Permalink | Reply to this

Re: Higher Gauge Theory and Elliptic Cohomology

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Here is what I find particularly interesting about the class of 2-reps which you are considering now:

for $G_{(2)}$ any strict 2-group (for those who don’t know it: it’s a groupoid with a certain monoidal structure on it) your 2-rep $\rho : \Sigma G_{(2)} \to (2\mathrm{Vect} = {}_{\mathrm{Vect}}\mathrm{Mod})$ is given on the 2-vector space $\rho(\bullet) = \mathrm{Func}(G_{(2)}, \mathrm{Vect}) \,.$ This

- is a 2-rep of $G_{(2)}$ by way of precomposing such functors with the action of $G_{(2)}$ on itself

- is a left $\mathrm{Vect}$ -module by way of postcomposition with the monoidal action of $\mathrm{Vect}$ on itself.

Remarkably, $\rho(\bullet)$ is nothing but the category of groupoid representations of the groupoid $G_{(2)}$ ! Of course.

So, if we are being pedantic and careful and write $\mathrm{Rep}(G_{(2)}) = \mathrm{Funct}(G_{(2)}, \mathrm{Vect})$ to be distinguished from $2\mathrm{Rep}(\Sigma G_{(2)}) = 2\mathrm{Funct}(\Sigma G_{(2)}, 2\mathrm{Vect}) \,,$ where the former is the category of groupoid reps of $G_{(2)}$ , while the latter is the 2-category of 2-reps of $G_{(2)}$ regarded as a 2-group, you choose the 2-rep whose 2-vector space is the category of groupoid reps: $\begin{aligned} &\rho \in 2\mathrm{Rep}(\Sigma G_{(2)}) \\ &\rho(\bullet) = \mathrm{Rep}(G_{(2)}) \end{aligned} \,.$ That looks like a nice idea, indeed!

But here it gets even more interesting, it seems:

as you go into some detail of explaining in your notes, the 2-group $G_{(2)} = \mathrm{String}_k(G)$ is special in that, as a groupoid, it is the bundle gerbe at level $k$ over the manifold $X = G$ underlying $G$ :

the surjective submersion is $Y = P G$ and the line bundle on $Y^{[2]} = PG \times_X P_G = \Omega G$ is nothing but the central extension $\hat \Omega_k G \to \Omega G \,.$ As for any line bundle gerbe, this bundle can be interpreted as a central extension of the groupoid $Y^{[2]} \stackrel{\to}{\to} Y \,.$ And this centrally extended groupoid is our $G_{(2)}$ . The 2-group structure on $G_{(2)}$ corresponds to the property of bundle gerbes people call “multiplicativity”.

(I am saying all this for readers who might not know.)

Anyway, the point is this: when we think of a bundle gerbe as being a groupoid, then a bundle gerbe module is essentially nothing but a representation of that groupoid!

A gerbe module on $G$ is nothing but a “twisted vector bundle” on $G$ , twisted by the gerbe. It’s the kind of structure that the Freed-Hopkins-Teleman theorem is all about.

So this means that – maybe up to the technicalities involving analysis which you mention – the 2-rep of $\mathrm{String}_k(G)$ which you consider is the category of gerbe modules for the canonical level- $k$ gerbe on $G$ .

In closing, a little addendum: in Freed-Hopkins-Teleman it is just the twisted bundles on $G$ which are equivariant under the adjoint action of $G$ on itself that appear. One might therefore wonder how this equivariance may be encoded in terms of constructions involving the 2-group $\mathrm{String}_k(G)$ . I think the answer is this:

the loop groupoid of $\Sigma \mathrm{String}_k(G)$ is $\Lambda \mathrm{String}_k(G) := 2\mathrm{Funct}( \Sigma \mathbb{Z} , \Sigma \mathrm{String}_k(G) )/_\sim \,,$ where I regard the group of integers as a 2-groupoid with a single object and no nontrivial 2-morphisms and consider all strict 2-functors of this into the 2-group $G_{(2)}$ regarded as a 1-object 2-groupoid, with pseudonatural transformations as morphisms and dividing out modifications.

This is supposed to be the Lie group analog of the central extension of the loop groupoid $\Lambda G = \mathrm{Funct}(\Sigma \mathbb{Z}, \Sigma G)$ for a finite group $G$ which Simon Willerton uses to tautologize FHT.

Reps of this groupoid (when the central extension is included) are like $G$ -equivariant twisted vector bundles on $G$ .

Similarly, one can see that – up to maybe some subtle analysis – reps of the loop groupoid $\Lambda \mathrm{String}_k(G)$ of the $\mathrm{String}_k(G)$ 2-group are $G$ -equivariant twisted vector bundles on $G$ , for $G$ a Lie group. This I describe in 2-Monoid of observables on $\mathrm{String}_k(G)$ .

Posted by: Urs Schreiber on August 1, 2007 10:22 AM | Permalink | Reply to this

Re: Higher Gauge Theory and Elliptic Cohomology

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Fascinating stuff. For a long time Danny Alissa and I had been thinking about this canonical representation in a more algebraic way, focusing on a discrete 2-group $\mathbf{G}$ , where we could ignore issues of analysis.

We spent a lot of time studying the canonical rep of $\mathbf{G}$ on its ‘function 2-algebra’, namely the category

(1)

\mathbf{G}^{Vect} = \{ functors from \; \mathbf{G} \; to Vect \}

This is a categorified version of the algebra of complex functions on a group $G$ , namely

(2)

G^{\mathbb{C}} = \{ functions from \; G \; to \mathbb{C} \} .

Just as $G^{\mathbb{C}}$ is a commutative algebra under pointwise multiplication, $\mathbf{G}^{Vect}$ is a `symmetric 2-algebra’.

Now, just as $G$ acts on the function algebra $G^{\mathbb{C}}$ in two really different ways — by translation and conjugation — $\mathbf{G}$ acts on the function 2-algebra $\mathbf{G}^{Vect}$ in two ways. We studied the translation action; that was our ‘canonical representation’. Are you hinting that we should focus on the conjugation action?

The translation action is nice because it gives the ‘regular representation’ of $\mathbf{G}$ , at least when $\mathbf{G}$ is finite. But the conjugation action should be ‘best’ in another sense!

After all, in the decategorified case, $G^{\mathbb{C}}$ is not just a commutative algebra — it’s a Hopf algebra, at least when $G$ is finite, so we can ignore issues of analysis. And the conjugation action of $G$ on $G^{\mathbb{C}}$ preserves this Hopf algebra structure. The left or right translation actions do not. So, returning to the categorified case, the conjugation action of $\mathbf{G}$ on $\mathbf{G}^{Vect}$ should preserve some sort of ‘Hopf 2-algebra’ structure, at least when $\mathbf{G}$ is finite.

Anyway, we focused on the case where $\mathbf{G}$ was finite! This let us ignore all issues of analysis (widely considered good to ignore), but also lulled us into ignoring the geometry (widely considered bad to ignore).

So, only later did we come back and ponder what $\mathbf{G}^\Vect$ should be like when $\mathbf{G}$ is our favorite 2-group, the fundamental gerbe of a compact simple Lie group $G$ .

Then I think Danny realized that $\mathbf{G}^\Vect$ — or some suitable analogue where you take the analysis into account — should be the category of twisted bundles on $G$ , twisted by the fundamental gerbe.

So yeah, the various representations of $\mathbf{G}$ on this should be very interesting! I guess you’re hinting that the objects in $\mathbf{G}^Vect$ which are ‘weak fixed points’ of the conjugation action should be the conjugation-equivariant twisted vector bundles over $G$ , which some of you guys seem to like so much — I’m never fully sure why.

And yeah, it should be fun to think about how $\mathbf{G}$ -2-connections give 2-connections on the associated vector 2-bundle with fiber $\mathbf{G}^{Vect}$ .

But now I see there are 2 really different ways to get $\mathbf{G}$ to act on $\mathbf{G}^{Vect}$ , giving different vector 2-bundles — and the conjugation way may be more interesting.

Posted by: John Baez on August 1, 2007 12:56 PM | Permalink | Reply to this

Re: Higher Gauge Theory and Elliptic Cohomology

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Are you hinting that we should focus on the conjugation action?

Well, not quite hinting. I don’t know which action would be the “right” one for the present purpose. It was just that after I arrived at noticing that your 2-rep is on the collection of twisted bundles, I felt like mentioning how equivariant twisted bundles happen to be related – without knowing if they are also relevant here.

But maybe they are. I’d need to think more about this.

‘weak fixed points’ of the conjugation action should be the conjugation-equivariant twisted vector bundles over $G$ , some of you guys seem to like so much — I’m never fully sure why.

If you ask me why these are important, I’ll say this:

$G$ -equivariant twisted vector bundles on $G$ are important, because these are the “2-states over the endpoints” of the string propagating on $\Sigma \mathrm{String}_k(G)$ .

That’s essentially Simon Willerton’s insight, but adopted for Lie groups and stated slightly more suggestively.

and the conjugation way may be more interesting.

Okay, now then, that’s getting very close to what I am rambling on about here for the last umpteen weeks:

Question: What is “group of conjugation actions” of a 2-group $\mathbf{G}$ on itself?

Answer: It’s the inner automorphism 3-group $\mathrm{INN}(\mathbf{G})$ .

I’d think it is a very good idea to consider this action. Since, as I mentioned above, we need to pass to $\mathrm{INN}(\mathrm{String}_k(G))$ anyway in order to be able to get all connections on our $\mathrm{String}_k(G)$ -bundle.

I’ll think about this.

Posted by: Urs Schreiber on August 1, 2007 1:31 PM | Permalink | Reply to this

Re: Higher Gauge Theory and Elliptic Cohomology

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One correction: I wrote

$G$ -equivariant twisted vector bundles on $G$ are important, because these are the “2-states over the endpoints” of the string propagating on $\Sigma \mathrm{String}_k(G)$ .

I think I made a mistake in dimension. It’s rather

$G$ -equivariant twisted vector bundles on $G$ are important, because these are the “3-states over the boundary” of the circular membrane propagating on $\Sigma \mathrm{String}_k(G)$ .

Posted by: Urs Schreiber on August 1, 2007 1:42 PM | Permalink | Reply to this

Re: Higher Gauge Theory and Elliptic Cohomology

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I’m way behind. What sort of entity is $\Sigma String_k(G)$ , and how does a string propagate on it?

You use $\Sigma G$ to mean the group $G$ regarded as a category (for me a group is a category), so is $\Sigma String_k(G)$ the 2-group $String_k(G)$ regarded as a 2-category? And if so, how many people know how to get a string to propagate on a 2-category???

Posted by: John Baez on August 1, 2007 2:04 PM | Permalink | Reply to this

Re: Higher Gauge Theory and Elliptic Cohomology

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What sort of entity is $\Sigma \mathrm{String}_k(G)$ , and how does a string propagate on it?

Using the idea of the “quantum charged $n$ -particle” # I can say, in principle, what it means for any $n$ -particle whose “parameter space” is an $n$ -category of sorts to propagate on any “target space” which is itself an $n$ -category.

Here one should think of the spaces of objects of these $n$ -categories as being the ordinary spaces one usually considers in physics, and the various morphisms are encoding various tangency relations between these objects.

So when I say something propagates on $\Sigma G_{(n)}$ it should really be thought of as something “propagating on the point” $\mathrm{pt}$ with a (possibly higher) orbifold structure on it.

For propagation on $\Sigma G_{(2)}$ I once discussed some more details in Globular Extended QFT of the Charged $n$ -Particle: String on BG (pdf).

You use $\Sigma G$ to mean the group $G$ regarded as a category (for me a group is a category)

At some point I realized that I find it very convenient to carefully distinguish by notation if, given a $k$ -tuply stabilized $n$ -category, or equivalently a $k$ -tuply monoidal $(n-k)$ -category, as what $j$ -category for $n-k \leq j \leq n$ I actually want to conceive it in the present context.

For instance there are important chains of inclusions of 2-categories like $\Sigma \Sigma \mathbb{C} \hookrightarrow \Sigma \mathrm{Vect} \hookrightarrow \mathrm{Bim} \hookrightarrow 2\mathrm{Vect}$ which would become very confusing if we omitted the $\Sigma$ s.

Or take the universal $G$ -bundle, in its groupoid incarnation, where it is the exact sequence $G \to \mathrm{INN}(G) \to \Sigma G$ of 1-groupoids. Here it is really crucial that we regard $G$ once as a 0-category with a monoidal structure (on the left) and once as a 1-object category (on the right). If I drop the $\Sigma$ s here, this becomes incomprehensible.

Similarly for the universal $G_{(2)}$ -2-bundle $G_{(2)} \to \mathrm{INN}(G_{(2)}) \to \Sigma G_{(2)} \,.$

And, I think, the very example which we are talking about is also a case in point:

there are at least two different kind of representations of a 2-group: one represents the 2-group regarded as a groupoid, the other as a 1-object 2-groupoid.

Anyway, I know it’s to some extent a matter of taste, but this is why I am using the $\Sigma$ s a lot.

so is $\Sigma \mathrm{String}_k(G)$ the 2-group $\Sigma \mathrm{String}_k(G)$ regarded as a 2-category?

Yes! A 1-object 2-groupoid.

And if so, how many people know how to get a string to propagate on a 2-category???

Few. Simon Willerton and Bruce Bartlett essentially know, though they might have a different way to put it.

Posted by: Urs Schreiber on August 1, 2007 2:26 PM | Permalink | Reply to this

Re: Higher Gauge Theory and Elliptic Cohomology

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I hope everyone remembers what a ‘weak fixed point’ of a group $G$ acting on a category $X$ is. It’s an object $x \in X$ together with isomorphisms $gx \stackrel{\sim}{\rightarrow} x$ for every $g \in G$ , such that whenever you have $g_1,g_2 \in G$ the two resulting isomorphisms from $(g_1 g_2)x$ to $x$ agree. There’s a pretty obvious category of weak fixed points of $G$ acting on $X$ .

However, I seem above to be discussing weak fixed points of a 2-group $\mathbf{G}$ acting on a category $\mathbf{G}^{Vect}$ . I haven’t thought about that concept as much, if at all.

Posted by: John Baez on August 1, 2007 2:10 PM | Permalink | Reply to this

Re: Higher Gauge Theory and Elliptic Cohomology

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I hope everyone remembers what a ‘weak fixed point’ of a group $G$ acting on a category $X$ is

By the way, if every object $x \in C$ is a weak fixed point, and if the morphisms connecting the acted-on objects to the original objects respect the orginal morphisms between these objects, then we have what I like to call a $G$ -flow on $C$ : it is a representation $\Sigma G \to \Sigma T_{\mathrm{Id}_C}(\mathrm{End}(C))$ of the group $G$ on the group of “flows” along $C$ , namely one transformation $\array{ & \nearrow \searrow^{\mathrm{Id}} \\ C &\Downarrow^\sim& C \\ & \searrow \nearrow_{g \cdot} }$ for each element $g \in G$ , such that this respects the product in $G$ under horizontal composition of transformations.

For instance if $C = P_1(X)$ is the path groupoid of some space $X$ , and $G = \mathbb{R}$ , this gives ordinary vector fields on $X$ , as we are discussing in Arrow-theoretic differential theory.

Posted by: Urs Schreiber on August 1, 2007 2:39 PM | Permalink | Reply to this

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July 31, 2007