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September 22, 2011

Division Algebras and Supersymmetry III

Posted by John Baez

guest post by John Huerta

Hi. Since this is my first ever post to the n-Café, let me introduce myself: I’m John Huerta, former student of John Baez, and now a postdoc at Australian National University. I was hired by Peter Bouwknegt, a string theorist who, like me, divides his time between the Departments of Mathematics and Theoretical Physics. In the six weeks that I’ve been here, I’ve already had a chance to see some of Australia’s amazing wildlife. I even got to feed kangaroos and wallabies! Here I am, offering some food to a wallaby:

John Huerta and friend.

I hope to see a lot more of Australia while I’m here! More seriously, I plan to do some work on T-duality and generalized geometry, some fascinating areas of mathematical physics on which Peter is an expert.

That’s enough about me. The real reason I am writing this post is to tell you about my first solo paper, which I just posted to the arXiv:

Abstract. Recent work applying higher gauge theory to the superstring has indicated the presence of ‘higher symmetry’. Infinitesimally, this is realized by a ‘Lie 2-superalgebra’ extending the Poincaré superalgebra in precisely the dimensions where the classical superstring makes sense: 3, 4, 6 and 10. In the previous paper in this series, we constructed this Lie 2-superalgebra using the normed division algebras. In this paper, we use an elegant geometric technique to integrate this Lie 2-superalgebra to a ‘Lie 2-supergroup’ extending the Poincaré supergroup in the same dimensions.

Briefly, a ‘Lie 2-superalgebra’ is a two-term chain complex with a bracket like a Lie superalgebra, but satisfying the Jacobi identity only up to chain homotopy. Simple examples of Lie 2-superalgebras arise from 3-cocycles on Lie superalgebras, and it is in this way that we constructed the Lie 2-superalgebra above. Because this 3-cocycle is supported on a nilpotent subalgebra, our geometric technique applies, and we obtain a Lie 2-supergroup integrating the Lie 2-superalgebra in the guise of a smooth 3-cocycle on the Poincaré supergroup.

I would love your comments on this paper. They would really help me to improve it! Below the fold, I’ll tell you what the paper is really about.

First, I should remind you about what we did in Division Algebras and Supersymmetry II: we used the normed division algebras to construct Lie 2-superalgebra extensions to the Poincaré superalgebra in spacetimes of the superstring dimensions: 3, 4, 6 and 10.

Wow! That’s already a lot to explain, but fortunately, John Baez already explained a lot of it here before:

So let me quote the inimitable John Baez:

The groups that physicists like all have Lie algebras. Lie algebras show up in particle physics because they describe how particles transform as they move around. But recently people have discovered gadgets called Lie 2-algebras, which do the same job for strings. And Lie 3-algebras, which do the same job for 2-dimensional membranes, usually called ‘2-branes’. And so on!

In fact, a Lie nn-algebra is actually a kind of hybrid structure: a blend of a Lie algebra and an nn-category.

But if you’re a practical sort of person, you may want to build a Lie nn-algebra starting from some stuff you can easily get your hands on. The simplest way is to start with a Lie algebra and a gizmo called an (n+1)(n+1)-cocycle: some sort of function satisfying some equation called a ‘cocycle condition’. From this, you can get a Lie nn-algebra that includes your original Lie algebra.

Cutting to the chase, this is exactly what do: we build a Lie 2-superalgebra extending the Poincaré superalgebra, using a 3-cocycle that exists exactly when the dimension of spacetime is 3, 4, 6 and 10. What is the Poincaré superalgebra, you ask? Its the ‘super version’ of the Poincaré algebra, of course! Quoting John one more time:

You see, special relativity says we live in Minkowski spacetime. The group of symmetries of Minkowski spacetime is called the Poincaré group. This has a Lie algebra: the Poincaré algebra. And there’s a supersymmetric analogue of all this, starting from ‘super-Minkowski spacetime’. Super-Minkowski spacetime unifies vectors and spinors in a nice way. And the supergroup of symmetries of super-Minkowski spacetime has a Lie superalgebra, called the Poincaré superalgebra.

Super!

So it isn’t too hard to believe that the Poincaré superalgebra is really important if you want to talk about supersymmetric theories of physics, like the superstring. Moreover, this 3-cocycle on the Poincaré superalgebra helps to make superstring theory tick: physicists explain how it is necessary for the superstring to have a kind of local supersymmetry called ‘Siegel symmetry’. I don’t really understand Siegel symmetry or why it’s needed, but I suspect that because it is local and super, it plays a big role in connecting the superstring to another famous theory with local supersymmetry: namely, supergravity.

The upshot is that we get a Lie 2-superalgebra extending Poincaré superalgebra whenever we can also have the superstring. Hence, we call this the superstring Lie 2-algebra, 𝔰𝔲𝔭𝔢𝔯𝔰𝔱𝔯𝔦𝔫𝔤\mathfrak{superstring}. And thanks to the work of super smart people like Hisham Sati, Urs Schreiber and Jim Stasheff on higher gauge theory:

we know this is no mere coincidence, but is turning out to be an important part of string theory.

I hope all of this convinces you that 𝔰𝔲𝔭𝔢𝔯𝔰𝔱𝔯𝔦𝔫𝔤\mathfrak{superstring} is important. But there’s something missing from this story: for gauge theory we have Lie groups and Lie algebras. For higher gauge theory, we have Lie 2-groups and Lie 2-algebras, and for higher gauge theory with supersymmetry, we have Lie 2-supergroups and Lie 2-superalgebras. What is the 2-supergroup that corresponds to the Lie 2-superalgebra 𝔰𝔲𝔭𝔢𝔯𝔰𝔱𝔯𝔦𝔫𝔤\mathfrak{superstring}?

In my new paper, I’ll tell you! And right now, I’ll tell you how I got the answer. Remember how we got our hands on 𝔰𝔲𝔭𝔢𝔯𝔰𝔱𝔯𝔦𝔫𝔤\mathfrak{superstring} using a 3-cocycle? Well, one can use 3-cocycles of a different sort to get our hands on 2-groups. And because this is the nn-Category Café, you probably won’t mind if I remind you how this works.

2-groups from 3-cocycles

Remember, a 2-group is a category 𝒢\mathcal{G} with invertible morphisms equipped with a multiplication functor:

m:𝒢×𝒢𝒢 m \colon \mathcal{G} \times \mathcal{G} \to \mathcal{G}

satisfying the group laws up to natural isomorphism. For instance, rather than being associative, we have an associator:

a g 1,g 2,g 3:(g 1g 2)g 3g 1(g 2g 3) a_{g_1,g_2,g_3} \colon (g_1 g_2)g_3 \to g_1(g_2g_3)

for any trio of objects g 1g_1, g 2g_2 and g 3g_3 in 𝒢\mathcal{G}. These natural isomorphisms then satisfy some laws of their own. For instance, the associator satisfies the pentagon identity, which says the following pentagon commutes:

Pentagon identity.

Suffice to say, every 2-group has a lot of data (objects, morphisms, natural isomorphisms) subject to several axioms. Luckily, thanks to a theorem of Joyal and Street, we can distill all this down to only four things:

  • a group GG,
  • an abelian group HH,
  • an action of GG on HH by automorphism,
  • an HH-valued 3-cocycle aa on GG.

This last item, the 3-cocycle, is just some function:

a:G 3H a \colon G^3 \to H

satisfying an equation called a ‘cocycle condition’. What is this equation? Well, it’s secretly just the pentagon identity! In fact, Joyal and Street proved that every 2-group is equivalent to a 2-group which has:

  • An object gg for each element of the group GG.
  • On each object gg, an automorphism h:ggh \colon g \to g for each element hh of the abelian group HH, and no morphisms between different objects. Composition of morphisms is addition in HH.
  • Multiplication of objects is multiplication in GG.
  • Multiplication of morphisms: h:gg,h:gg h \colon g \to g , \quad h' \colon g' \to g' is addition twisted by the action of GG: h+gh:gggg. h + g h' \colon g g' \to g g' .
  • The associator: a g 1,g 2,g 3:(g 1g 2)g 3g 1(g 2g 3) a_{g_1,g_2,g_3} \colon (g_1 g_2)g_3 \to g_1(g_2g_3) given by the 3-cocycle a:G 3Ha \colon G^3 \to H.

This works out to be a 2-group, because aa satisfies the pentagon identity if and only if it is a 3-cocycle in the more mundane sense of group cohomology. Note that even though multplication is associative (GG is a group, after all), we can still have nontrivial associators. Weird! In fact, we can think of aa as kind of obstruction to making a 2-group strictly associative, as spelled out in HDA5.

For this story, however, what matters is that:

  • A 3-cocycle on a group GG gives a 2-group extending GG.

And this is an example of a broad pattern:

  • A 3-cocycle on a Lie group GG gives a Lie 2-group extending GG.
  • A 3-cocycle on a supergroup GG gives a 2-supergroup extending GG.

In fact, we saw this pattern with Lie algebras, too:

  • A 3-cocycle on a Lie algebra 𝔤\mathfrak{g} gives a Lie 2-algebra extending 𝔤\mathfrak{g}.
  • A 3-cocycle on a Lie superalgebra 𝔤\mathfrak{g} gives a Lie 2-superalgebra extending 𝔤\mathfrak{g}.

After all, this was how we got our hands on the Lie 2-superalgebra, 𝔰𝔲𝔭𝔢𝔯𝔰𝔱𝔯𝔦𝔫𝔤\mathfrak{superstring}, which extends the Poincaré superalgebra.

Now, of course, I haven’t defined ‘2-supergroups’, or for that matter Lie 2-algebras. Right now, the exact definition is not important. What’s important is that 3-cocycles give examples. Because that’s how we are going to integrate 𝔰𝔲𝔭𝔢𝔯𝔰𝔱𝔯𝔦𝔫𝔤\mathfrak{superstring}: we will find a 3-cocycle on the Poincaré supergroup which integrates the 3-cocycle used to define 𝔰𝔲𝔭𝔢𝔯𝔰𝔱𝔯𝔦𝔫𝔤\mathfrak{superstring}.

Integrating cocycles

Let me finish off with a taste of how this integration works, but made easier because it’s not super: I’ll tell you how to integrate a 3-cocycle on a Lie algebra, 𝔤\mathfrak{g}, to a 3-cocycle on the corresponding Lie group, GG. Of course, that means I’ll have to tell you what each of those things is, so let me give you two definitions that suit my purpose:

  • A 3-cocycle on the Lie algebra 𝔤\mathfrak{g} is a closed, left-invariant 3-form on the Lie group GG.

Sneaky, eh? I’m already working with something defined on the Lie group. But I’m not cheating, because we can always take a structure defined on the Lie algebra and left translate it to get a structure defined on the group.

  • A 3-cocycle on the Lie group GG is a smooth function a:G 3a \colon G^3 \to \mathbb{R} satisfying the equation: a(g 2,g 3,g 4)a(g 1g 2,g 3,g 4)+a(g 1,g 2g 3,g 4)a(g 1,g 2,g 3g 4)+a(g 1,g 2,g 3)=0. a(g_2, g_3, g_4) - a(g_1 g_2, g_3, g_4) + a(g_1, g_2 g_3, g_4) - a(g_1, g_2, g_3 g_4) + a(g_1, g_2, g_3) = 0 .

At this point, you’re probably wondering where that silly equation came from. Remember the pentagon identity:

Pentagon identity.

where composition is just addition, and you’re not far from the equation above. But I have carefully arranged the terms hopefully to make my punchline, a surprise application of Stokes’ theorem, as easy as possible.

So, given a Lie algebra 3-cocycle ω\omega (a closed, left-invariant 3-form), how are we going to get a Lie group 3-cocycle? Just integrate our 3-form over a 3-simplex:

a(g 1,g 2,g 3)= [1,g 1,g 1g 2,g 1g 2g 3]ω. \displaystyle a(g_1, g_2, g_3) = \int_{[1,g_1,g_1g_2,g_1g_2g_3]} \omega .

Here [1,g 1,g 1g 2,g 1g 2g 3][1,g_1,g_1g_2,g_1g_2g_3] is some standard way of filling out a 3-simplex in GG with the given vertices. Choose this ‘standard filling’ carefully, and you can also make sure it has the virtue of being left-invariant:

g[g 0,,g p]=[gg 0,,gg p] g[g_0, \ldots, g_p] = [g g_0, \ldots, g g_p]

so let’s assume that here. Constructing these left-invariant simplices is the biggest challenge of the paper, and requires some technical assumptions about GG—namely, that the exponential map:

exp:𝔤G \exp \colon \mathfrak{g} \to G

be a diffeomorphism. When this happens, GG is called exponential.

Once this simplex is constructed, however, we’re home free. I’ll leave the last bit of work, checking that aa is a 3-cocycle, to you. I can’t resist giving a big hint, however: use Stokes’ theorem!

Posted at September 22, 2011 9:26 AM UTC

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14 Comments & 0 Trackbacks

Re: Division Algebras and Supersymmetry III

Hi John! Nice to meet you in Adelaide.

Just a brief (possibly stupid) question: the exponential map is only a diffeomorphism if the manifold underlying the Lie group is Euclidean space (hence contractible). Do you mean ‘local diffeomorphism’?

I know there are some tricky things once one leaves the world of ordinary finite-dimensional (compact?) Lie groups.

Can you clarify?

Posted by: David Roberts on September 23, 2011 1:08 AM | Permalink | Reply to this

Re: Division Algebras and Supersymmetry III

Wow, you really do mean diffeomorphism. I would have thought this a huge constraint, but I never knew that simply connected Lie groups integrating nilpotent Lie algebras were exponential.

Posted by: David Roberts on September 23, 2011 5:56 AM | Permalink | Reply to this

Re: Division Algebras and Supersymmetry III

Hi David. Likewise, great meeting you in Adelaide.

Yes, I really mean a diffeomorphism. And thanks to the fact that all simply-connected nilpotent Lie groups are exponential, there is no shortage of examples, such as the Heisenberg group.

You may be wondering how this is relevant, though. After all, the Poincaré supergroup is not nilpotent, and there’s no way it’s diffeomorphic to a super vector space. I mean, just look at it:

SISO(V)=Spin(V)T. \mathrm{SISO}(V) = \mathrm{Spin}(V) \ltimes T .

That’s the Poincaré supergroup: the semidirect product of the Lorentz group Spin(V)\mathrm{Spin}(V) and a supergroup I call the ‘supertranslation group’, TT, which you should think of as the group of ‘translation symmetries’ of Minkowski superspacetime. Ignoring TT for the moment, that Spin(V)\mathrm{Spin}(V) is going to keep this group from being topologically trivial, which seems to foil my plan to integrate the 3-cocycle.

Ah, but I didn’t mention that the Lie superalgebra 3-cocycle I’m trying to integrate is supported on 𝔗\mathfrak{T}, the Lie superalgebra of TT. And TT, unlike the full Poincaré supergroup, is nilpotent. So, we can integrate the cocycle there.

So, this gives me a 3-cocycle on TT, one which turns out to be Lorentz invariant. And that invariance lets me extend the 3-cocycle to all of the Poincaré supergroup, solving the problem.

Posted by: John Huerta on September 23, 2011 8:12 AM | Permalink | Reply to this

Re: Division Algebras and Supersymmetry III

Hi John,

if I have read your thesis, which parts of this preprint should I pay attention to? By the way, it still says “in this thesis” at two places in arXiv:1109.3574: on p. 6 and on p. 26.

Here are some comments on the blog entry.

You write:

So it isn’t too hard to believe that the Poincaré superalgebra is really important if you want to talk about supersymmetric theories of physics,

I know that you are intending to make it expositional, but I find this a bit misleading. Because supersymmetry is defined to be about the Poincaré superalgebra. So what does it mean to say that it is plausible that it is related to the Poincaré superalgebra?

physicists explain how it is necessary for the superstring to have a kind of local supersymmetry called ‘Siegel symmetry’.

Here I think you are mixing up things. What is called Siegel symmetry or more often kappa-symmetry is a certain redundancy in the action Green-Schwarz action functional . This is a manifestly target-space supersymmetric alternative to the manifestly worldsheet supersymmetric NSR-type action. While this makes the GS-formulation attractive, its quantization is difficult. All the relation of string structures to quantum anomalies that your work is related to comes from the worldsheet susy formulation. For your purposes you can entirely ignore “Siegel symmetry” as a technical issue in a different setup.

Instead, it seems you want to explain why

this 3-cocycle on the Poincaré superalgebra helps to make superstring theory tick

To be precise, in view of the above, this 3-cocycle is about what makes the heterotic string tick. The heterotic string has only half of the worldsheet spinors that the closed type II superstring has, therefore the fermionic piece of its path integral is a kind of square root of that of the type II string. This kind-of-square-root has, as all square roots, a sign ambiguity. In general this cannot be consistently chosen globally: in general the heterotic sigma model has a fermionic quantum anomaly.

Whatever makes this quantum anomaly go away is what “makes the heterotic string tick”. For the case that the background Yang-Mills field of the heterotic string is trivial, this condition is precisely that the target space has string structure, in that the first fractional Pontryagin class 12p 1\frac{1}{2}p_1 vanishes. As one can show, the bosonic 3-cocycle on 𝔰𝔬\mathfrak{so} is precisely the infinitesimal version of this class, in that it Lie integrates to this class. The fermionic 3-cocycle that you are dealing with is kind of the super-partner of this fermionic 3-cocycle.

You write:

every 2-group has a lot of data […] Luckily, thanks to a theorem of Joyal and Street, we can distill all this down to only four things:

I have two comments on this. First, while you speak of a 2-group as a group-like category, it is in fact even a group-like groupoid. The classification of group-like-groupoids is much older than the classification of monoidal categories. After delooping, this is the classification of pointed, connected homotopy 2-types, which is ancient knowledge.

A similar comment would apply to where you appeal to tricategories in your thesis. Since all your tricategories are actually tri-groupoids aka homotopy 3-types, appealing to the full theory of tricategories here is, while nice, a bit of an overkill, and maybe tends to hide to the reader that there is a lot of standard knowledge about these objects.

But maybe that’s a matter of taste. My second comment on this is a bit more substantial: in either case, what these classification results give is the classification of bare 2-groupoids, ignoring smooth or topological or supergeometric or any other cohesive structure. In other words, this is is about classification of discrete 2-groups.

It requires a bit more discussion, I think, to see that – and how – this classification lifts to cohesive 2-groups.

One can start by observing that every smooth or supergeometric 2-group is presentable by a presheaf of discrete strict 2-groups. This is corollary 2.3.2 here. Then apply the usual classification in the presheaf topos. Then stackify.

Or one can observe that the classification is really just the Postnikov data in an infinity-category applied to the delooped 2-group as an object in the ambient cohesive \infty-topos.

In either case, one finds a classification that looks like that for discrete 2-groups, but only if we pass from (super)Lie 2-groups to the much larger class of (super)smooth 2-groups (including diffeological 2-groups and also non-concrete smooth 2-groups).

Posted by: Urs Schreiber on September 24, 2011 1:13 AM | Permalink | Reply to this

Re: Division Algebras and Supersymmetry III

Hi Urs,

If you’ve read my thesis, you won’t learn anything new from this preprint. I am just portioning my thesis off for publication as part of the “Division Algebras and Supersymmetry” series. Naturally, that’s why the word “thesis” shows up; thanks for catching it!

You write:

supersymmetry is defined to be about the Poincaré superalgebra.

True enough, at least at the Lie superalgebra level. I even say this somewhere in this paper, when I introduce the Poincaré superalgebra. I was hoping my conversational style of writing would be welcome to people not in the know.

Speaking of not being in the know, let’s tackle my own ignorance:

While this makes the GS-formulation attractive, its quantization is difficult. All the relation of string structures to quantum anomalies that your work is related to comes from the worldsheet susy formulation. For your purposes you can entirely ignore “Siegel symmetry” as a technical issue in a different setup.

So far, my work has been purely classical, beginning with an attempt to relate the division algebras to the supersymmetry of the GS superstring. According to Green-Schwarz-Witten, the WZW term in the GS action is supersymmetric if and only if my favorite 3-cocycle exists. That term, in turn, gives the action Siegel symmetry, which cuts the fermionic degress of freedom in half, but also forces the background to be supergravity, at least according to this guy:

So we could say the WZW term is needed for Siegel symmetry, but I’m not sure why Siegel symmetry is needed—the reason given seems to be that we want to cut the fermionic degrees of freedom in half, but why? So what if we have a theory with twice as many fermions as bosons? It looks like a perfectly legitimate, supersymmetric Lagrangian to me… This is where my ignorance starts to kick in.

Whatever makes this quantum anomaly go away is what “makes the heterotic string tick”. For the case that the background Yang-Mills field of the heterotic string is trivial, this condition is precisely that the target space has string structure, in that the first fractional Pontryagin class 12p 1\frac{1}{2}p_1 vanishes. As one can show, the bosonic 3-cocycle on 𝔰𝔬\mathfrak{so} is precisely the infinitesimal version of this class, in that it Lie integrates to this class. The fermionic 3-cocycle that you are dealing with is kind of the super-partner of this fermionic 3-cocycle.

This is fascinating stuff. I have heard you say some of this before, and I’m glad you keep bringing me back to it. But I just pored over some of the literature on the heterotic string, and I can’t find the 3-cocycle anywhere. Where is it, and how do you know that it is a kind of “fermionic partner” of the canonical 3-cocycle on 𝔰𝔬\mathfrak{so}?

As for your remarks regarding Joyal–Street, you’re right that this is ancient knowledge from the perspective of homotopy 2-types. Any lack of clarity on this point comes from me being more comfortable with a 2-group than a 2-type. Thanks to Tim Porter, I know that 2-types were classified in terms of crossed modules by Whitehead many decades ago. But when did people realize that they were also classified by 3-cocycles? And when did weak 2-groups get into the game? I could look some of this stuff up pretty easily, but I wasn’t worried about doing it for a blog post. In my paper, it would merit a remark, but I don’t emphasize classification there.

Indeed, I don’t know how to classify smooth 2-groups. I only prove that one can obtain examples from smooth 3-cocycles, not that every smooth 2-group is equivalent to such data. In fact, I point out that using van Est’s definition of smooth Lie group cohomology is surely inadequate in general. I also point out that Baez–Lauda Lie 2-groups are also known to fall short in important respects, and the state-of-the-art is to be found in your work.

I’d love to understand that all, and I have finally started reading your habilitation. But I have lots of other things to do, so it may take a while.

Of course, working on a project together could help.

In any case, thanks for the comments. I appreciate it.

Posted by: John Huerta on September 26, 2011 3:37 PM | Permalink | Reply to this

Re: Division Algebras and Supersymmetry III

Hi John,

oh, okay, I see. I thought you were talking about why one needs to extend by the cocycles. But you just meant the existence of the cocycles. Sure, that guarantees the proper spacetime supersymmetry.

So far, my work has been purely classical,

The reason for considering the extensions classified by these cocycles, as you do, hence for considering the String-2-group, or the Fivebrane-6-group, etc, comes from quantization, along the lines I indicated.

Of course, working on a project together could help.

Any time. We have more than enough projects waiting for manpower. I think I had suggested some projects to you earlier, but if these weren’t to your tastes, we’ll find something else. Let’s talk by email.

Indeed, I don’t know how to classify smooth 2-groups.

But I told you in my previous message!

Posted by: Urs Schreiber on September 27, 2011 3:00 AM | Permalink | Reply to this

Re: Division Algebras and Supersymmetry III

This is good. So far, my understanding for why one should extend to the Lie 2-superalgebra or 2-supergroup has been mostly a moral one—like a mountain climber, “because it’s there.” Or, really, something like:

Moral reason: The 3-cocycles are there, so the extensions to a 2-thing are also automatically there. Sati, Schreiber and Stasheff know how to actually use ‘em.

But now you’re telling me what the use really is: tackling quantum anomalies. Great! I’ve been interested in this stuff since Branislav Jurčo started explaining anomaly cancellation with 2-gerbes to me at Oberwolfach.

Nonetheless, it is still true that the 3-cocycle in question shows up in the decidedly nonquantum dimensions of 3, 4, and 6. I suspect that’s also good for something, and I bet our combined references on kappa symmetry are quietly expressing the answer: they’re there for supergravity.

As for how to classify smooth 2-groups:

But I told you in my previous message!

I noted this, but I didn’t understand what you said. Let’s take your two answers in turn:

One can start by observing that every smooth or supergeometric 2-group is presentable by a presheaf of discrete strict 2-groups. This is corollary 2.3.2 here. Then apply the usual classification in the presheaf topos. Then stackify.

That’s a classification in three steps, each one of which I find utterly mysterious. I have a vague inkling that “presentable by a presheaf of discrete strict 2-groups” is related to famous thing I know by the name of the “functor of points”. That is, if we have a space that is presented by a functor (AKA, a presheaf) on some site:

G:Site opSet G \colon Site^{\op} \to Set

and that space also happens to be a group-object in the category of set-valued contravariant functors (AKA, presheaves) on SiteSite, then over every object in xSitex \in Site, we get an ordinary, discrete group, defined on the set G(x)SetG(x) \in Set.

For anyone reading this who doesn’t know why this is the case, let me quickly explain, just as a Baez acolyte should. GG being a group in the category of presheaves means that I am quietly equipping it with natural transformations called multiplication, inversion and identity-assignment:

m:G×GG m \colon G \times G \Rightarrow G inv:GG inv \colon G \Rightarrow G id:1G id \colon 1 \Rightarrow G

which satisfy the usual group axioms. Here, 1 is the presheaf that takes everybody in SiteSite to your favorite one-element set in SetSet.

But these natural transformations have components for each object in xSitex \in Site:

m x:G(x)×G(x)G(x) m_x \colon G(x) \times G(x) \to G(x) inv x:G(x)G(x) inv_x \colon G(x) \to G(x) id x:1G(x) id_x \colon 1 \to G(x)

and now these are just functions between sets. What’s more they also satisfy the group axioms, precisely because mm, invinv and idid did. They thus make the plain ol’ set G(x)G(x) into a group.

So, we might as well think of GG as a group-valued presheaf in the first place:

G:Site opGrp. G \colon Site^{\op} \to Grp .

Is this at all the kind of thing you mean? I looked at the actual corollary you linked to, but it didn’t give any sort of intuition like that.

As for the other two steps in this classification:

Then apply the usual classification in the presheaf topos. Then stackify.

Both of these are mysterious right now, but I bet I could learn them in a short span of time. They also sound worth learning. But even when I have sorted out the technical details, I am already worried that I won’t know what this sketch of a classification really means, in a moral sense. Such is the stumbling block of a moralist like me.

As for your other classification sketch:

Or one can observe that the classification is really just the Postnikov data in an infinity-category applied to the delooped 2-group as an object in the ambient cohesive \infty-topos.

On the surface, this sounds a lot closer to the moral reason, just based on the fact that some topologists have told me the construction from 3-cocycles is just “one step in the Postnikov tower”. Come to think of it, I’ve seen this sort of thing before:

But now the learning curve is presented by the phrase “the delooped 2-group as an object in the ambient cohesive \infty-topos”. Lots of adjectives, and infinitely categorified to boot! I’m just not sure where to start with this one, besides your habilitation.

So that’s why I said I still don’t know how to classify smooth 2-groups. Perhaps I should have added, “for now”.

Posted by: John Huerta on September 27, 2011 11:39 AM | Permalink | Reply to this

Re: Division Algebras and Supersymmetry III

By the way, just so we’re clear, I used the word “moral” in two different ways in the above comment. The first was for my reason for studying 3-cocycles (though this was not the only reason). The second was for a deep down mathematical understanding of the classification of smooth 2-groups.

Posted by: John Huerta on September 27, 2011 11:56 AM | Permalink | Reply to this

Re: Division Algebras and Supersymmetry III

@John H:But now the learning curve is presented by the phrase “the delooped 2-group as an object in the ambient cohesive infty-topos”. Lots of adjectives, and infinitely categorified to boot!

My sympathies! Stating things in maximal generality can totally obscure the meaning in terms of more accessible `plain English’ examples ;-)

Posted by: jim stasheff on September 27, 2011 1:20 PM | Permalink | Reply to this

Re: Division Algebras and Supersymmetry III

Sorry for being quiet for so long. I’d enjoy discussing more, but simply didn’t and don’t have the time right now. I am visiting Pittsburgh and am working with Hisham on some projects and have been giving talks the last three days. Somehow that absorbs time.

Since the talks seemed to have been well received, I reworked my notes for them into the motivation section of the cohesive-thesis.

Maybe you find the new section 1.1 Motivation / 1.1.2 From quantum anomaly cancellation useful.

Incidentally, the last talk I gave an hour back was at CMU to the Homotopy Type Theory group. I tried to highlight the fact that the simple abstract axioms for cohesive \infty-toposes induce rich structure that knows about differential geometry and differential cohomology. That seems remarkable, because the chances to explain to, say, Coq what a local and \infty-connected \infty-topos are seem to be much higher than to directly explain to it what even just, say, a cotangent bundle is.

The notes for that talk I now made 1.1 Motivation / 1.1.3 From higher topos theory.

(But I need to go through this again tonight with a bit more leisure.)

Concerning your (John Huerta’s) other questions about how to think of sheaves; these are meant to be answered in the section 1. Introduction / 1.1. General abstract theory. If you want to do me a great favor, we’d go through that section and you’d tell me wherever you get stuck. This is the section where I want to be pedagogical, not in the technical section 2 that I pointed you to previously.

Posted by: Urs Schreiber on September 30, 2011 9:31 PM | Permalink | Reply to this

Re: Division Algebras and Supersymmetry III

I’m not sure why Siegel symmetry is needed

A geometric (supergeometric) explanation of the meaning of κ\kappa-symmetry is discussed in

I.N. McArthur, Kappa-Symmetry of Green-Schwarz Actions in Coset Superspaces (hep-th/9908045)

and in the references listed there. It is all about the super-Cartan geometry of the super-target space (super-Poincaré mod super-rotations).

Posted by: Urs Schreiber on September 27, 2011 4:37 AM | Permalink | Reply to this

Re: Division Algebras and Supersymmetry III

For more on the gemetric interpretation of κ\kappa-symmetry see also

  • Joaquim Gomis, Kiyoshi Kamimura, Peter West, Diffeomorphism, kappa transformations and the theory of non-linear realisations (arXiv:hep-th/0607104),

especially section 3.

Posted by: Urs Schreiber on September 27, 2011 4:50 AM | Permalink | Reply to this

Re: Division Algebras and Supersymmetry III

Hi John,

I’ll be joining you at ANU next year. I’ve also been spending lots of time thinking about higher categories, so I’m sure we’ll have lots of fun talking about all this.

best,
Scott Morrison

Posted by: Scott on September 25, 2011 5:53 AM | Permalink | Reply to this

Re: Division Algebras and Supersymmetry III

Hi Scott! Welcome to the ANU family. The math department is quite lively, and I really like it.

I remember when you gave a cool talk on blob homology at Riverside. I look forward to talking more when you get here.

Posted by: John Huerta on September 26, 2011 3:46 PM | Permalink | Reply to this

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