August 28, 2006

10D SuGra 2-Connection

Posted by Urs Schreiber

We have seen ($\to$) that 11-dimensional supergravity is a gauge theory of a 3-connection, taking values in a certain Lie 3-algebra, $\mathrm{sugra}_{11}$, which is an extension of the super-Poincaré-1-algebra by a 4-cocycle.

I claimed ($\to$) that non-fake flat $n$-connections with values in an $n$-algebra $g$ are to be interpreted in terms of their curvatures, which are flat $(n+1)$-connections with values in

(1)$\mathrm{inn}(g) \,,$

the $(n+1)$-algebra of inner derivations of $g$, where flatness encodes the $n$-Bianchi identity.

The task is hence to compute $\mathrm{inn}(\mathrm{sugra}_11)$.

That’s straightforward, but pretty hard. I am still hoping to figure out a shortcut, computing $\mathrm{inn}(g)$ directly at the level of FDAs. But I don’t see the pattern yet.

Meanwhile, it might be a good idea to study related but simpler examples. As John has already mentioned ($\to$) it might be easier to look at 10-dimensional supergravity first.

Here I present a discussion of what should be the bosonic part of the 2-connection governing 10-dimensional supergravity. The main point is to understand, from a categorical point of view, the relation

(2)$H = d B + \mathrm{CS}(A)$

between the curvature 3-form $H$, the Kalb-Ramond 2-form $B$ and the $\mathrm{SO}(32) \oplus \mathrm{SO}(9,1)$-connection $A$, which governs the Green-Schwarz anomaly cancellation ($\to$) (Phys.Lett.B149:117-122,1984, ($\to$)).

Following Killingback ($\to$, $\to$), we expect 10-dimensional supergravity to be governed by the 2-group $\mathrm{String}_G$ ($\to$, $\to$), where $G$ is the the 10-dimensional Lorentz group times an internal factor $\mathrm{SO}(32)$ or $E_8\times E_8$.

The Lie-2-algebra of this 2-group has a weak skeletal incarnation which is, as noticed first by André Henriques ($\to$) nothing but the Baez-Crans Lie 2-algebra $g_\hbar$ (see example 50 of Alissa Crans’ thesis). The Koszul-dual FDA of this guy is particularly simple:

Let $g$ be some Lie algebra, and let $h = \mathrm{Lie}(\mathbb{R})$. On the free graded-commutative algebra

(1)$\wedge^\bullet( g^* \otimes h^*)$

with $g^*$ in degree 1 and $h^*$ in degree 2, we define a differential of grade 1 in terms of a basis $\{a^a\}$ of $g^*$ and $\{b\}$ of $h^*$ by setting

(2)\begin{aligned} & d a^a + \frac{1}{2}C^a{}_{bc}a^b a^c = 0 \\ & d b + \frac{1}{6}C_{abc}a^a a^b a^c = 0 \,. \end{aligned}

Here $C^a{}_{bc}$ are the structure constants of $g$ in the chosen basis and $C_{abc} = k_{aa'}C^a{}_{bc}$, where $k$ is the Killing form of $g$.

This $d$ is nilpotent due to the Jacobi identity in $g$.

We would like to find the Lie 3-algebra $\mathrm{inn}(g_\hbar)$ of inner derivations of this Lie 2-algebra.

Instead of trying to strictly derive this, I’ll notice that due to various constraints there is not much of a choice and an obvious ansatz will do the job.

From the study of $\mathrm{inn}(h\to g)$ for $(h\to g)$ an arbitrary strict Lie-2-algebra (example 12 of the FDA Lab ($\to$)), we expect the algebra of $\mathrm{inn}(g_\hbar)$ to be based on a vector space consisting of two copies of $g_\hbar$, one of which shifted by one in degree. So consider

(3)$\bigwedge^\bullet (g^* \otimes (g^* \otimes h^*) \otimes h^*)$

with the first $g^*$ in degree 1, the expression in brackets in degree 2 and the last $h^*$ in degree 3.

Let $\{a^a\}$ be a basis of $g^*$ in degree 1, $\{b\}$ a basis of $h^*$ in degree 2, $\{r^a\}$ a basis of $g^*$ in degree 2 and $\{c\}$ a basis of $h^*$ in degree 3.

We know (example 12 of the FDA Lab ($\to$)) that the 2-algebra of inner derivations of $g$ itself is encoded in the differential defined by

(4)\begin{aligned} & d a^a + \frac{1}{2}C^a{}_{bc}a^b a^c + r^a = 0 \\ & d r^a + \frac{1}{6}C^a{}_{bc}a^b r^c = 0 \end{aligned} \,.

All available modifications of the original $d b + C_{abc}a^a a^b a^c = 0$ are hence given by

(5)$d b + \frac{1}{6}C_{abc}a^a a^b a^c + u \, k_{ab} a^a r^b - c = 0 \,,$

for $u \in \mathbb{R}$ some real parameter. For the particular choice $u = 1$ we find that $d$ is nilpotent precisely if

(6)$dc + k_{ab}r^a r^b = 0 \,.$

In summary, we have the FDA on $\bigwedge^\bullet (g^* \otimes (g^* \otimes h^*) \otimes h^*)$ determined by

(7)\begin{aligned} & d a^a + \frac{1}{2}C^a{}_{bc}a^b a^c + r^a = 0 \\ & d r^a + C^a{}_{bc}a^b r^c = 0 \\ & d b + \frac{1}{6}C_{abc}a^a a^b a^c + k_{ab} a^a r^b - c = 0 \\ & d c + k_{ab}r^a r^b = 0 \,. \end{aligned}

I dare to call this FDA $\mathrm{inn}(g_\hbar)$, though I have only shown that it is one particular choice from a 1-parameter collection of admissable ansätze.

Given this, it is easy to derive the degrees of freedom of a flat (curvature-)$3$-connection with values in this Lie 3-algebra.

First of all, this is a degree-preserving map $\Phi$ from $\bigwedge^\bullet (g^* \otimes (g^* \otimes h^*) \otimes h^*)$ to the deRham complex of our base manifold, compatible with the algebra structure. In other words, we have a $g$-valued 1-form

(8)$\Phi(a^a) = A^a \,,$

a $g$-valued 2-form

(9)$\Phi(r^a) = -F^a$

and an ordinary 2-form

(10)$\Phi(b) = B \,,$

and finally a 3-form

(11)$\Phi(c) = H \,.$

In order for this to be a chain map, we need to have $[Q,\Phi] = 0$, where $Q = d_{\mathrm{inn}(g_\hbar)} \oplus d_{\text{deRham}}$. This condition is equivalent to requiring that

(12)$F = d A + A\wedge A$

is the curvature of $A$ and that

(13)$H = d B + \mathrm{CS}(A)$

is the 3-form curvature, where $\mathrm{CS}(A)$ denotes the Chern-Simons 3-form ($\to$) of $A$.

The last condition - the Bianchi-identity on $H$ - is then automatic

(14)$d H \propto \mathrm{tr}(F\wedge F) \,.$

This is exactly the bosonic field content of the gauge sector that we expect.

In closing, I note that the above should essentially be the local infinitesimal version of a connection on the String-gerbe which Danny Stevenson describes in section 6 of his notes. Notice how in these notes, too, the String gerbe is governed by a flat CS-2-gerbe.

You can find notes on the computations involved here in the FDA Laboratory.

Posted at August 28, 2006 2:15 PM UTC

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Re: 10D SuGra 2-Connection

The progress you’re making is really impressive!

Let’s try to figure out the Lie 3-algebra $inn(g)$ for a Lie 2-algebra g without resorting to any guesses. I’m sure you’re on the right track, but it would be nice to have a systematic approach, especially for when we categorify.

The only way I can imagine is fairly strenuous. You may have already tried it. If $g$ is the Lie 2-algebra of a Lie 2-group $G$, $inn(g)$ should be the Lie 3-algebra of the Lie 3-group $Inn(G)$ - the Lie 3-group of inner automorphisms of $G$. I think we can, at least with sufficient energy, figure out the definition of $Inn(G)$ and then work out its Lie 3-algebra $inn(g)$. In fact, $inn(g)$ should only depend on $g$; we should be able to see how. Then, we can use this as a definition of $inn(g)$ for any Lie 2-algebra $g$, regardless of whether it has a corresponding Lie 2-group.

(In fact any Lie n-algebra $g$ will come from a Lie n-group in the sense of Henriques . For the present purposes, this simply amounts to saying we can locally integrate the Lie n-algebra to a Lie n-group. In fact Henriques goes further, but we don’t need to worry about that here.)

Before we dive into this, a question: why are you sure we need $inn(g)$ instead of the potentially larger $der(g)$?

There’s something funny about $inn(g)$, after all. It should be analogous to $Inn(G)$. When $G$ is a group, the 2-group $Inn(G)$ is equivalent to the trivial 2-group, right? I assume you mean to define this 2-group using a crossed module $t: G \to G$ with $t = 1$. So, all objects in this 2-group are uniquely isomorphic. So, it’s equivalent to the trivial 2-group. By analogy, I expect that for any Lie algebra $g$, the Lie 2-algebra $inn(g)$ should be equivalent to the trivial Lie 2-algebra. And, categorifying this idea rather blindly, I’d guess that for any Lie n-algebra $g$, the Lie (n+1)-algebra $inn(g)$ is equivalent to the trivial one!

Can we really get something interesting with a $inn(g)$-connection when $inn(g)$ is equivalent to something trivial?

I guess it’s easiest to check this for n = 1, and I assume you already have. If you could explain what’s really going on here, that would be great. You may recall a similar discussion over on David’s blog, where I was attacking the idea of Klein 2-geometry being interesting when the relevant 2-group is equivalent to a trivial one.

(Could there be a difference between 2-groups and Lie 2-groups here??? A smooth category with all objects uniquely isomorphic may not be smoothly equivalent to the trivial category. But in fact, I don’t think that’s a way out - I think $Inn(G)$ is even smoothly equivalent to the trivial 2-group.)

Anyway, if we wanted to compute $der(g)$, I’d first ponder $Aut(G)$ and then differentiate. If $G$ is a 2-group, $Aut(G)$ is the 3-group with

• equivalences $G \to G$ as objects,
• pseudonatural transformations between these as morphisms,
• modifications between these as 3-morphisms.

Sitting inside here should be $Inn(G)$, but I’m not quite sure how to define it.

Posted by: John Baez on August 31, 2006 1:43 PM | Permalink | Reply to this

Re: 10D SuGra 2-Connection

You may have already tried it. If $g$ is the Lie 2-algebra of a Lie 2-group $G$, $\mathrm{inn}(g)$ should be the Lie 3-algebra of the Lie 3-group $\mathrm{Inn}(G)$ - the Lie 3-group of inner automorphisms of $G$. I think we can, at least with sufficient energy, figure out the definition of $\mathrm{Inn}(G)$ and then work out its Lie 3-algebra $\mathrm{inn}(g)$

Yes. I did precisely this, for the case where $G_2$ is a strict 2-group.

(Let me write $G_2$ for 2-groups.)

For $G_2$ strict, I (define and) compute $\mathrm{Inn}(G_2)$ in the first part of this document (which belongs to the $n$-curvature entry ($\to$), where some background information is provided).

The result is that $\mathrm{Inn}(G_2)$ is almost like a crossed module $((H \to G)\to G)$, where $H \to G$ is the crossed module defining $G_2$. So it’s something like a crossed module of crossed modules, as one would expect.

For that reason, it is not hard to read off the differential version of $\mathrm{Inn}(G_2)$. While I haven’t yet tried to write down a detailed proof, I believe one can easily see that the result is that which is encoded in the FDA which I discuss in example 12 of my FDA Laboratory.

So for strict 2-groups $G_2$ I think I do know $\mathrm{Inn}(G_2)$ as well as $\mathrm{inn}(g_2)$.

Good. The issue with the above entry was that I tried to work with the skeletal weak incarnation of $\mathrm{string}_g$.

Of couse, we know that this is equivalent to a strict Fré Lie-2-group. So I could just use that.

And it’s true, I could. It should be equivalent to what I am doing above.

But I have not yet quite understood how, under that equivalence, the nice curvature relation

(1)$H = dB + \mathrm{CS}(A)$

translates.

It’s almost clear, but not quite. Our strict string 2-connection involves a 1-form $A$ with values in the path algebra of $g$ and a 2-form $B = (B_\mathrm{cent},B_\mathrm{loop})$ with values in the centrally extended loop algebra, and should have curvature

(2)$H = d_A B = (dB_\mathrm{cent} +\int\langle A , B'_\mathrm{loop}\rangle, H_\mathrm{loop}) \,.$

Clearly, the central term here wants to translate into the term $dB + \mathrm{CS}(A)$. But I don’t quite see it yet.

Therefore my interest in the weak Baez-Crans version of $\mathrm{string}_g$, for which I only made the above educated guess of what $\mathrm{inn}(\mathrm{string}_g)$ must look like.

Maybe I should recap how I did that: $\mathrm{string}_g$ is like the strict $(u(1) \stackrel{\text{triv}}{\to} g)$, with a nontrivial associator thrown in. So I take the FDA of the strict $\mathrm{inn}(u(1)\to g)$, which I do understand, and throw the associator term in.

Then it turns out that using the available tensors and obeying all the grading constraints and everything, there is precisely only one further term which can be added to the differential of the FDA. So the FDA of the weak $\mathrm{inn}(\mathrm{string}_g)$ must be like that of the strict case, but incorporating the associator term and that unique extra term. What I show above is that throwing this term in does indeed give a consistent FDA, which does imply the curvature relations found in the heterotic string.

But, as you say, this is not the most satisfying procedure and I would eventually want to derive $\mathrm{inn}(\mathrm{string}_g)$ rigorously.

why are you sure we need inn(g) instead of the potentially larger der(g)?

That’s just because I am looking at smooth - hence continuous - 2-functors from 2-paths to $\mathrm{Aut}(G_2)$. They will only hit the component connected to the identity. I think that’s $\mathrm{Inn}(G_2)$. But maybe I am wrong.

When $G$ is a group, the 2-group $\mathrm{Inn}(G)$ is equivalent to the trivial 2-group, right?

[…]

Can we really get something interesting with a $\mathrm{inn}(g)$-connection when $\mathrm{inn}(g)$ is equivalent to something trivial?

[…]

You may recall a similar discussion over on David’s blog, where I was attacking the idea of Klein 2-geometry being interesting when the relevant 2-group is equivalent to a trivial one.

Yes, indeed. In that discussion I did in fact go off-topic repeatedly by talking about these $(G\to G)$-2-connections.

So, yes $G \to G$ is equivalent to $1\to 1$, even smoothly, I think. I derive this by regarding both beasts as 2-categories, take the only possible 2-functors between them and notice that

(3)$(\bullet \stackrel{g}{\to} \bullet) \;\; \mapsto \;\; \array{ \bullet &\stackrel{g}{\to}& \bullet \\ \mathrm{Id}\downarrow\;\; &\Downarrow g^{-1}& \;\;\downarrow \mathrm{Id} \\ \bullet &\stackrel{\mathrm{Id}}{\to}& \bullet }$

is the pseudonatural transformation that makes these two 2-functors weak inverses of each other.

Now consider a smooth 2-functor from 2-paths to $\Sigma(G\to G)$. We know it is given by a $\mathrm{Lie}(G)$ 1-form $A$ and - by fake flatness - the 2-form $B = -F_A$.

In fact, this 2-functor is flat. Trivial - you might say. Because $d_A B = - d_A F_A = 0$.

So flatness/triviality of this 2-functor encodes precisely the Bianchi identity of a non-flat/non-trivial 1-functor with values in $G$.

I believe there is a general principle behind this, which I try to describe in the “$n$-curvature” text linked here.

Applying this principle to $G_2$-2-transport than lead me to find non-fake flat 2-transport in terms of flat (“trivial”) 3-transport.

You might want to reformulate this as follows:

A trivial $(n+1)$-gerbe is trivialized by a nontrivial $n$-gerbe.

Reading this the other way around, we may associate to a nontrivial $n$-gerbe the trivial $(n+1)$-gerbe which it trivializes. This $(n+1)$-gerbe we could call the curvature gerbe of the original gerbe.

Take the simplest example of line bundle gerbes with connection. If these are trivial (better: trivializable) and flat, they have a globally defined 2-form $B$, and indeed this is nothing but the curvature 2-form of the line bundle with connection which trivializes the gerbe. The flatness of the trivial flat gerbe is precisely the Bianchi identity of the connection on the non-trivial line bundle.

So in abstract terms, it seems what is going on is this:

We have something which is trivializable. Then the nontrivial information is the precise choice of trivializing morphism we use.

A bundle with connection is a morphism, which identitfies a trivializable gerbe with the comletely trivial gerbe.

And so on.

This is currently my way of looking at things. As you know, Danny arrives at what should be the same conclusion, coming from a complementary perspective. But I don’t know if we may talk about that in public. He said he is trying to finish some notes on this stuff this week.

I guess it’s easiest to check this for $n = 1$, and I assume you already have.

You can find more details on these toy examples in the example section at the end of the $n$-curvature text ($\to$).

$\mathrm{Aut}(G)$ is the 3-group with equivalences $G\to G$ as objects, pseudonatural transformations between these as morphisms, modifications between these as 3-morphisms.

Yes, that’s precisely is what I compute in the first section here.

Posted by: urs on August 31, 2006 3:04 PM | Permalink | Reply to this

Re: 10D SuGra 2-Connection

I wrote:

It’s almost clear, but not quite.

Now I am beginning to understand it.

The thing is, the equivalence from the big, strict string algebra to the small, weak one respects composition only weakly. The failure is measured by the compositor $\phi_2$ in our lemma 33.

This means, that if we compose our transport 2-functor from 2-paths to the big, strict string group with the map to the small, weak thing (we can do that for sufficiently small paths), it will have nontrivial compositor, too.

But this implies that we have to be careful when defining its curvature, because there is now a slight ambiguity as to what the boundary of a 3-path is, for instance. Do we compose first and transport then, or the other way around.

This ambiguity is fixed by inserting the compositor morphsims everywhere to switch between the two possibilities. This then modifies what you would naively obtain for the curvature, by these extra contributions.

A comparison with lemma 33 and example 12 in the FDA lab then shows that the contribution of these compositors indeed does involve the term $\langle A(2\pi), F_A(2\pi)\rangle$, where $A$ is the connection with values in the path algebra of $G$ and $A(2\pi)$ denotes it value at the endpoint.

I have yet to write it down in full detail, but it seems plausible that this is how we get the expected $H = dB_\mathrm{cent} + \mathrm{CS}(A(2\pi))$ also from transport with values in the big,strict string group, for which we understand $\mathrm{Inn}(\mathrm{String}_G)$ a little better. (The $\langle A\wedge A \wedge A\rangle$ component in $\mathrm{CS}(A)$ comes from the associator. That’s the easy part.)

I can’t work anymore on that today. Maybe tomorrow.

Posted by: urs on August 31, 2006 6:48 PM | Permalink | Reply to this

Re: 10D SuGra 2-Connection

I’m being sort of a dilettante at this game, at least so far. I’d like to get more involved, since this is most fascinating combination of ideas I’ve seen for a long time. But you’re so far ahead of me it’s hard to keep up! It’s a lot of work to read as fast as you can write.

So, instead of trying to first learn everything you’ve already done, I’ll continue my tactic of thinking out loud about various questions, many of which you’ve already answered somewhere. I hope this isn’t too annoying.

For $G_2$ strict, I (define and) compute Inn$(G_2)$ in the first part of this document […]

The result is that Inn$(G_2)$ is almost like a crossed module $(H \to G) \to G$, where $H \to G$ is the crossed module defining $G_2$. So it’s something like a crossed module of crossed modules, as one would expect.

I know you’re not mainly interested in strict 2-groups (until you bring the loop groups into the game), but let me think about this. First I’ll define this concept of “inner automorphism 3-group of a strict 2-group” in an elegant but overly strict way. Then I’ll do it in a better way.

Overly Strict Inner Automorphisms

Start one level down. We have a functor

$\mathrm{Inn} : \mathrm{Grp} \to 2\mathrm{Grp}$

sending each group to its 2-group of inner automorphisms - where right now all my n-groups are strict. This functor can be written down very explicitly. It’s “algebraic” in a certain technical sense, so it works not just for groups but for group objects in any category with finite products.

So, we can internalize it in Cat! A group in Cat is a 2-group, and a 2-group in Cat is a 3-group. So, we get

$\mathrm{Inn} : 2\mathrm{Grp} \to 3\mathrm{Grp}$

(and so on for higher n). It’s all quite slick. If we’re fond of crossed modules, we can use the equivalence of categories

$2\mathrm{Grp} \simeq \mathrm{CrossMod}$

$\mathrm{Inn} : \mathrm{Grp} \to \mathrm{CrossMod}$

This simply sends $G$ to the crossed module you denote in a rough way by

$1: G \to G.$

Again this functor is algebraic, so we can internalize it in any category with finite products, in particular Cat, and get

$\mathrm{Inn} : 2\mathrm{Grp} \to 2\mathrm{CrossMod}$

where $2\mathrm{CrossMod}$ is my notation for crossed modules internal to Cat.

In fact, these so-called categorical crossed modules have been studied here:

apparently even for less strict ones than I’m talking about now! (Everything I’m doing here is totally strict.)

So, if $G_2$ is a 2-group, we obtain a crossed module of 2-groups called Inn$(G_2)$, which we denote in a rough way by:

$1: G_2 \to G_2$

We can also think of this as a “crossed module of crossed modules” instead of “crossed modules of 2-groups”, since

$2\mathrm{Grp} \simeq \mathrm{CrossMod}$

Before I mention the bad news, I’ll just add that Aaron Lauda and I internalized the above equivalence in Prop. 32 of HDA5, and it works in any category K with finite products such that the category of group objects in K has finite limits. That extra italicized fine print is required because we need to take a kernel when going from a 2-group to its crossed module. But, all this fine print hold when K = Set or even K = Diff.

A Better Kind of Inner Automorphisms

Alas, all the above stuff is too strict for you, since in your writeup you define Inn($G_2$) using pseudonatural transformations instead of natural ones - and surely for very good reasons.

So, instead of getting an honest crossed module of crossed modules, perhaps you really do get something that’s only like a crossed module of crossed modules - presumably with some laws weakened.

Are some laws weakened? Maybe you can tell me.

If so, perhaps what you’re getting can still be seen as a categorical crossed module in the sense of Vitale et al.

Of course, there’s a lot of jargon being thrown around here, but ultimately my question amounts to this: is your Inn($G_2$) a strict 3-group or a somewhat weak one?

Hmm…

A strict 3-group is a special sort of strict monoidal 2-category. Now I’m hoping you’re getting a semistrict monoidal 2-category.

Hey - now I seem to be answering my own question! If you take any 2-category $C$ and look at

• strict 2-functors from $C$ to $C$,
• pseudonatural transformations between these,
• modifications between these

we get something called End($C$) which is a semistrict monoidal 2-category, because it’s sitting inside the semistrict 3-category Gray, and it has one object $C$.

(Remember, Gray is the semistrict 3-category built from

• 2-categories,
• strict 2-functors between these,
• pseudonatural transformations between these,
• modifications between these

It was discovered by John Gray. Australian category theorists use the term Gray-category instead of “semistrict 3-category”, because such an entity is really just a category enriched over Gray. Similarly, they use Gray-monoid instead of semistrict monoidal 2-category.)

When $C$ is actually our 2-group $G_2$, we can find inside the semistrict monoidal 2-category End($C$) a smaller one called Aut($G_2$), built from

• invertible strict 2-functors from $C$ to $C$,
• pseudonatural isomorphisms between these,
• invertible modifications between these

Because this is a semistrict monoidal 2-category where everything is (strictly) invertible, this deserves to be called a semistrict 3-group.

Sitting inside Aut($G_2$) there’s a smaller semistrict 3-group called Inn($G_2$), which you define.

Now I think I understand things fairly well - but I won’t delete all the junk at the beginning of this post where I was doing everything in an overly strict way. Someone, somewhere, may care someday. The universe is big, and it will last a long time.

It would be nice if these semistrict 3-groups corresponded to Vitale et al’s categorical crossed modules. These semistrict 3-groups are, by the way, “sufficiently general” - every weak 3-group is equivalent to one of these!

Posted by: John Baez on September 1, 2006 4:42 AM | Permalink | Reply to this

Re: 10D SuGra 2-Connection

I’ll continue my tactic of thinking out loud about various questions, many of which you’ve already answered somewhere.

That’s great. In particular, since you will mention all those questions which I failed to address anywhere.

You answer most of your questions yourself. But I will still comment on some of them.

Are some laws weakened? Maybe you can tell me.

Yes. That’s why I emphasized last time that I only almost get a crossed module of crossed module.

One of the two consistency laws of the would-be crossed module (of crossed modules) turns out to fail. But it fails only up to an isomodification.

I discuss this on p. 5 of my notes. And I believe I identify the reason for this weakening exactly as you do below, too, even though I don’t use the term “Gray category”.

If so, perhaps what you’re getting can still be seen as a categorical crossed module in the sense of Vitale et al.

That’s what I expect. In Vienna (I, II) a couple of people were talking to me about crossed squares, 3-groups, crossed modules of crossed modules and the like. I was told to look at Conduché’s work on 3-groups. But I must admit to my shame that I still haven’t.

is your $\mathrm{Inn}(G_2)$ a strict 3-group or a somewhat weak one?

Yes, it’s slightly weak. I think everything is strict except for the exchange law for 2-morphisms. But since, as I emphasize in my notes, there is a unique 3-morphisms between any two parallel 2-morphisms, the weakness is rather mild.

Now I’m hoping you’re getting a semistrict monoidal 2-category.

You are answering this yourself below to the positive. But please remind me: what precisely is the distinction between generally weak and semistrict? (I should know that, but I seem to have forgotten.)

Remember, Gray is the semistrict 3-category built from 2-categories, strict 2-functors between these, pseudonatural transformations between these, modifications between these

I didn’t remember that, even though Zoran Skoda once was so nice to send me some of Gray’s texts. But yes, that’s precisely what I am looking it. In fact, I never really say so explicitly, but I do indeed restrict attention to strict 2-functors. But then allow pseudonatural transformations between them.

Posted by: urs on September 1, 2006 11:50 AM | Permalink | Reply to this

Re: 10D SuGra 2-Connection

Urs writes:

Please remind me: what precisely is the distinction between generally weak and semistrict?

In general there are three strategies to categorification: strict, weak, and semistrict.

When we categorify a definition strictly, all equational laws are still maintained as equations. When we categorify it weakly, all equational laws are replaced by equivalences satisfying coherence laws up to equivalences satisfying coherence laws… and so on. When we categorify it semistrictly, we try to keep as many equations as possible while still being able to prove a coherence theorem saying any “weak” gadget of the type being defined is equivalent to one of our “semistrict” ones.

The strict route is easy but insufficiently general. The weak route is hard but sufficiently general and ultimately best. The semistrict route is a kind of careful compromise. There’s no clear philosophy yet of semistrict categorification - it’s a bit like trying to put your hand as near to the candle flame of strictness as possible without getting burned! It’s not even true that there’s always a unique way to do this!

Some examples will help before I finally tackle the example you’re interested in.

Strict monoidal categories are actually “semistrict”, because Mac Lane’s theorem says every weak monoidal category is equivalent to a strict one.

A better example comes from braided monoidal categories. Every weak braided monoidal category is equivalent to one where the associator and left/right unit laws are equations (= identity morphisms). But, we can’t simultaneously demand that the braiding be an equation - that would be too strict! So, in a semistrict braided monoidal category, everything is an equation except the braiding. (Just to confuse us, people often call these “strict”.)

A strict monoidal 2-category is a strict 3-category with one object. Equivalently, it’s a monoid object in the monoidal category 2Cat. All laws hold on the nose, as equations!

A weak monoidal 2-category or monoidal bicategory is a tricategory with one object. Everything about it is as weak as possible, so we have an associator which satisfies the pentagon equation only up to a 2-isomorphism called the pentagonator which satisfies some equations of its own… and so on.

A semistrict monoidal 2-category is a Gray-category with one object, or equivalent, a monoid object in Gray. Unravelling what this means, you’ll see everything is as strict as possible except that the bifunctoriality of the tensor product of 1-morphisms, namely this equation

$(f \otimes 1)(1 \otimes g) = (1 \otimes g)(f \otimes 1)$

familiar in monoidal categories, has been replaced by a 2-isomorphism called the tensorator:

$\otimes_{f,g}: (f \otimes 1)(1 \otimes g) \Rightarrow (1 \otimes g)(f \otimes 1)$

which satisfies certain coherence laws of its own.

The big coherence theorem of Gordon, Power and Street says that every tricategory is equivalent to a Gray-category. The proof also implies that every weak monoidal 2-category is equivalent to a semistrict one.

For this and other reasons, I’m pretty sure that semistrict 3-groups are sufficiently general while still being pretty easy to deal with - for masters of the universe like you, that is.

But, things will get a lot more scary when we move from string theory to M-theory and start wanting “semistrict 4-groups”. Nobody except perhaps Sjoerd Crans seems to understand what semistrict 4-category is! Just as Gray defined a tensor product of 2-categories so that categories enriched over these would be semistrict 3-categories, Sjoerd Crans has defined a tensor product of semistrict 3-categories so that categories enriched over these would be semistrict 4-categories! But, I’ve never met anyone except him who claims to understand his work.

So, before we climb this high on the dimensional ladder, it will be good to switch over to Lie n-algebras and guess some general recipe for figuring out the Lie (n+1)-algebra $inn(g)$ from the Lie n-algebra $g$, without taking a detour through n-groups!

Nonetheless, the lowly case of string theory is already interesting, and it’ll be great fun seeing it from all different viewpoints. Ross Street will be so excited when he hears that Gray-categories are relevant to physics!

(By the way, if you ever want to learn about the Gray tensor product of 2-categories and the resulting concept of semistrict monoidal 2-category, you’ll probably find the beginning of my paper with Martin Neuchl to be the easiest reference. Gray’s papers are pretty hard.)

Posted by: John Baez on September 1, 2006 1:12 PM | Permalink | Reply to this

Re: 10D SuGra 2-Connection

So, before we climb this high on the dimensional ladder, it will be good to switch over to Lie n-algebras and guess some general recipe for figuring out the Lie (n+1)-algebra $\mathrm{inn}(g)$ from the Lie $n$-algebra $g$, without taking a detour through $n$-groups!

Yes, agreed.

Given an ordinary algebra $A$, I can regard the subset of its elements of the form $\mathrm{exp}(a)$, for $a \in A$. This subset is no longer an algebra. But it is a group.

I think we should restrict attention to groups that arise this way for the moment.

So I want to categorify this.

I want to start with an $n$-algebra $A$ and form the $n$-group $\mathrm{exp}(A)$, simply by restricting attention to those $n$-morphsims in $A$ which are of the form

(1)$1 + a + \frac{1}{2}a \cdot a + \cdots \,,$

where $\cdot$ is the product functor $A \times A \to A$.

The point is that this allows us to seamlessly pass back and forth between computations involving $n$-groups and their Lie $n$-algebras, simply by using Baker-Campbell-Hausdorff and truncation of the exponential series.

So given a Lie $n$-algebra $L$, it should be possible to form the universal enveloping $n$-algebra $U_L$ from it in the standard way. Our $n$-group then is $\exp(U_L)$.

I guess I am just saying what it means to integrate a Lie $n$-algebra locally. But this is enough for most of our purposes of $n$-connections.

Maybe what I say here is more than obvious. My point is that this is the right way to study the sort of questions that we are faced with here. I will try to type some more notes in order to illustrate what I have in mind.

But please let me know if the above program sounds way too naive.

Posted by: urs on September 1, 2006 1:53 PM | Permalink | Reply to this

Re: 10D SuGra 2-Connection

Urs wrote:

Given an ordinary algebra $A$, I can regard the subset of its elements of the form $\mathrm{exp}(a)$, for $a \in A$. This subset is no longer an algebra. But it is a group.

I’m not sure if that’s true. For example, in $\mathrm{SL}(2,\mathbb{R})$ the exponential map is not onto, and its range is not a subgroup.

(But this isn’t coming from an algebra, so it’s not a counterexample to your claim.)

I think we should restrict attention to groups that arise this way for the moment.

The point is that this allows us to seamlessly pass back and forth between computations involving $n$-groups and their Lie $n$-algebras, simply by using Baker-Campbell-Hausdorff and truncation of the exponential series.

I have the feeling this tactic is exactly what Ezra Getzler was advocating here.

It gets a bit scary when $\mathrm{exp}$ isn’t 1-1, since then the same group element has lots of names.

So, it’s nicest for Lie groups where the exponential map is 1-1 and onto. Alas, there aren’t too many Lie groups like this: the most famous ones are those whose Lie algebras are nilpotent. (For example, any Lie algebra of strictly upper triangular matrices.) This is why Getzler was restricting attention to nilpotent Lie $n$-algebras in some of his work.

But, if you can figure out formulas using this trick, and the formulas apply to all Lie $n$-algebras, I am happy to take them as definitions!

Posted by: John Baez on September 3, 2006 6:27 AM | Permalink | Reply to this

Re: 10D SuGra 2-Connection

John wrote:

Why are you sure we need $inn(g)$ instead of the potentially larger $der(g)$?

Urs wrote:

That’s just because I am looking at smooth - hence continuous - 2-functors from 2-paths to Aut$(G_2)$. They will only hit the component connected to the identity. I think that’s Inn$(G_2)$. But maybe I am wrong.

Well, for simple Lie groups $G$ the usual automorphism group $Aut(G)$ has the inner automorphism group $Inn(G)$ as its identity component. But, this isn’t true for other Lie groups, like $\mathbb{R}^2$: any linear transformation of the plane is an automorphism of $\mathbb{R}^2$, but only the identity is inner.

Of course this isn’t an example you care about, and it’s only a tiny step on the way towards understanding inner automorphism (n+1)-groups of n-groups, but it makes a point: the concept of being “inner” is conceptually very different from being “connected to the identity by a continuous path”.

So, I’m not yet convinced that you really want to think about $inn(g)$. In fact, seeing how the whole automorphism 2-group of a group is so important in the theory of nonabelian gerbes, I might guess that $der(g)$ is what you really want.

It might not matter in some examples you’re considering. It’s a small point. But it would be nice to know what’s really going on here.

Posted by: John Baez on September 1, 2006 1:28 PM | Permalink | Reply to this

Re: 10D SuGra 2-Connection

But it would be nice to know what’s really going on here.

I agree.

Yes, maybe in the end we really want all of $\mathrm{Aut}(G_2)$. What I have tried to show so far is just that using merely $\mathrm{Inn}(G_2)$ already produces the degrees of freedom that one needs for certain applications.

There is one constraint, I believe. As I describe in my curvature notes (html, pdf), one crucial condition on the target $(n+1)$-group for my $n$-transport is that it has a unique $(n+1)$-morphisms between any pair of parallel $n$-morphisms.

Another thing we should eventully be more careful about regarding our terminology, is that, for instance when I write

(1)$\mathrm{Inn}(G) = (G \stackrel{\mathrm{Id}}{\to} G)$

I do distinguish inner automorphisms $\mathrm{Ad}_g$ and $\mathrm{Ad}_{g'}$ that differ by an element of the center.

So, more properly, I should possibly write

(2)$(G \stackrel{\mathrm{Id}}{\to} G) = \mathrm{Inn}(G) \,\times\, (Z(G)\stackrel{\mathrm{Id}}{\to} Z(G)) \,.$
Posted by: urs on September 1, 2006 2:14 PM | Permalink | Reply to this

Re: 10D SuGra 2-Connection

Urs writes:

There is one constraint, I believe. As I describe in my curvature notes (html, pdf), one crucial condition on the target $\left(n+1\right)$-group for my $n$-transport is that it has a unique $(n+1)$-morphism between any pair of parallel $n$-morphisms.

Let’s take $n = 1$. If the constraint you mention is really, really necessary, then $\mathrm{AUT}(G)$ can’t be the 2-group you want, since this does not always satisfy that constraint.

To get a strict 2-group that satisfies that constraint, it is necessary and sufficient that the target map $t : H \to G$ in the corresponding crossed module be 1-1 and onto. Right?

If so, the obvious candidate is indeed to take $G = H$, $t = 1$, and let the action of $G$ on $H$ be via conjugation.

The 2-group corresponding to this crossed module probably does not deserve the name $\mathrm{INN}(G)$, because it’s not always a subcategory (in the usual naive sense) of $\mathrm{AUT}(G)$.

(I’m using all capital expressions for 2-groups here, to distinguish them from plain old groups with similar names.)

If I think about higher gauge theory a bit, it seems clear that we do need the crossed module $1: G \to G$. We turn a connection into a fake-flat 2-connection simply by defining $B = F$, so we’re just taking $G = H$ and $t = 1$.

We’re not using the crossed module $G \to \mathrm{Aut}(G)$ or even the crossed module $G \to \mathrm{Inn}(G)$.

We must not be fooled by the fact that in some cases all 3 of these crossed modules agree!

If so, this suggests that we should not be thinking about “AUT” or “INN” in the higher-dimensional cases, either! Instead, something different that just happens to be the same in some cases.

Does this seem persuasive?

Posted by: John Baez on September 4, 2006 3:04 AM | Permalink | Reply to this

Re: 10D SuGra 2-Connection

Does this seem persuasive?

Yes, indeed.

I had similar thoughts. In my discussion on non-fake flat transport I start by computing what I call $\mathrm{Inn}(G_2)$, then demonstrate that it happens to have unique 3-morphisms and then conclude that it is hence suitable for my purposes.

I fully agree that, while something is clearly right with the idea that we want to pass from $G_2$ to “something like” $\mathrm{Inn}(G_2)$, I might not yet fully understand the principle that is at work here.

Posted by: urs on September 4, 2006 5:15 PM | Permalink | Reply to this

Re: 10D SuGra 2-Connection

might not yet fully understand

It must be staring into our faces.

It seems that what is to be understood is the integrated version of what Danny Stevenson describes here (in the context of your Chicago lectures), which is a categorified - and differentiated - application of the basic principle of higher Schreier theory.

We want the non-differentiated version of this!

Given a principal bundle $P$, its Atiyah sequence is (more details on p. 4 of Danny’s notes)

(1)$0 \to \mathrm{ad}(P) \to T P/G \to T M \to 0 \,,$

or, taking sections,

(2)$0 \to \Gamma(\mathrm{ad}(P)) \to \Gamma(T P/G) \to \Gamma(T M) \to 0 \,.$

This should in fact correspond to a sequence of Lie algebroids, I think.

A conection on $P$ is a splitting of this sequence - if we allow it to forget the algebroid structure. By Schreier theory, we may remember the algebroid structure and say that the connection is instead a morphism $\Gamma(T M) \to \mathrm{DER}(\Gamma(\mathrm{ad}(P)))$

What we need to do is to reformulate this piece-by-piece while passing from Lie algebroids to Lie groupoids.

What is the corresponding sequence

(3)$A \to B \to C$

of Lie groupoids?

By the reasoning from p.8 of Danny’s notes, a connection should be a functor $C \to \mathrm{Aut}(A)$ - and that should be where the $\mathrm{Aut}$ comes in.

I am way too tired to think straight, but I do want to drop this comment.

Clearly, $C$ must be some flavor of a path groupoid $P(X)$ of base space $X$.

Ordinarily, we take $B$ to be the transport groupoid of $P$, $B = \mathrm{Trans}(P) = P\times P/G$ and say a connection is a smooth functor

(4)$\mathrm{tra} : P(X) \to \mathrm{Trans}(P) \,.$

This, and its categorification, is what we understand well. But it leads to fake flatness.

So now we want to slightly reformulate this, such that it behaves “better” under categorification.

So I guess we have to take $B$ to be an extension of the path groupoid by $\mathrm{Trans}(E)$. Take objects of $B$ to be points in $X$ and morphisms to be pairs $(\gamma,f)$ with $\gamma$ (the class of) a path in $X$ and $f$ a morphism between the fibers over the endpoints.

Then what is $A$? Looks like $A$ must be $\mathrm{End}(P)$, regarded as a skeletal groupoid, i.e. objects are points of $x$ and morphisms $x \to x$ are endomorphisms of the fiber over $x$. The morphism

(5)$A \to B$

then takes such an endomorphism $P_x \stackrel{g}{\to} P_x$ and sends it to the pair containing the constant path

(6)$(P_x \stackrel{g}{\to} P_x) \mapsto (\gamma_x : \sigma \mapsto x, g) \,.$

Unless I am really too tired, this makes

(7)$\mathrm{End}(P) \to B \to P(X)$

really an exact sequence of groupoids.

So, by generalized Schreier theory, we should now be looking for smooth functors

(8)$P(X) \to \mathrm{Aut}(\mathrm{End}(P))$

Restricting attention for the moment to the case where our bundle $P$ is trivial, we have $\mathrm{End}(P) \simeq X \times G$ and the above functor is nothing but a functor

(9)$P(X) \to \mathrm{Aut}(G) \,.$

And we want to categorify that.

Hm, I’d say even if $P$ is nontrivial we have…

No, I won’t say anymore today. I am too tired. But this looks like the way to go to me.

Posted by: urs on September 4, 2006 10:13 PM | Permalink | Reply to this

Re: 10D SuGra 2-Connection

I’ll refine some of the things I said above.

First of all, the Lie groupoid corresponding to the Lie algebroid $T X$ is, at least when $X$ is simply connected, the pair groupoid of $X$. In general it should be the fundamental groupoid.

So let me write

(1)$P(X)$

for the fundamental groupoid of $X$, objects of which are points in $x$ and morphisms are homotopy classes of paths (not thin homotopy classes, full homotopy classes).

For patches $U \subset X$ diffeomorphic to $\mathbb{R}^n$ a $G$-valued connection is indeed a pseudofunctor

(2)$P(U) \to \mathrm{Inn}(G) \subset \mathrm{Aut}(G)$

which assigns $P \exp(\int_{\path_{x_0}^{x_1}} A) \in G$ to pairs of points $(x_0,x_1)$, for $\path_{x_0}^{x_1}$ any fixed path between these two points.

This doesn’t respect composition in $P(U)$, but the failure is precisely measured by the curvature of $A$ and we indeed get a pseudofunctor.

That’s a slightly different setup than the one where we use thin homotopy classes of paths and a proper 1-functor. But it’s what we need here. And there is an obvious translation between the two descriptions.

Next, I want to give a better description of the groupoid I called $\mathrm{End}(P)$.

First of all I should rename the bundle $P$ to $B$, say. Otherwise there are too many $P$s in the game.

So let $B$ be a G-bundle over $X$.

Morphisms in the groupoid $\mathrm{End}(B)$ here were supposed to be torsor morphisms $B_x \to B_x$. This set of morphisms is canonically identitfied with the group $G$ itself. Moreover, the action of $G$ on the fiber $B_x$ translates into an adjoint action on $\mathrm{Hom}(B_x,B_x)$.

Hence, I think what I called $\mathrm{End}(B)$ can equivalently be called

(3)$\mathrm{Ad}(B) \,,$

which is the bundle of groups obtained by associating $G$ to the bundle $B$ by means of the adjoint action of $G$ on itself.

Just for notational simplicity, let me assume for the moment that $X$ is simply connected. Then the exact sequence of groupoids that I am talking about here is

(4)$\mathrm{Ad}(B) \to \mathrm{Trans}(B) \to P(X) \,.$

Now, this is beginning to look good. This sequence of groupoids has good chances of giving the Atiyah sequence

(5)$0 \to \mathrm{ad}(B) \to TB/G \to T X \to 0$

of Lie algebroids when differentiated.

So it seems we should be looking for pseudofunctors

(6)$\mathrm{tra} : P(X) \to \mathrm{Aut}(\mathrm{Ad}(B)) \,.$

As I indicated above, locally over $U$, where $B$ is trivializable this does reproduce correctly a $G$-connection.

But I am a little puzzled about how this works globally, because $\mathrm{Aut}(\mathrm{Ad}(B))$ has only a single object. That’s strange.

Posted by: urs on September 5, 2006 12:50 PM | Permalink | Reply to this

Re: 10D SuGra 2-Connection

I wrote:

That’s strange.

In fact, I think I need to clarify for myself what I want to mean by $\mathrm{Aut}(\mathrm{Ad}(B))$.

I believe what we really want here is not the automorphisms 2-group of the groupoid, but the 2-groupoid whose

1) objects are the End-sets of $\mathrm{Ad}(B)$;

2) morphisms are isofunctors between these;

3) 2-morphisms are natural transformations between these functors.

Posted by: urs on September 5, 2006 1:24 PM | Permalink | Reply to this
Read the post n-Transport and Higher Schreier Theory
Weblog: The n-Category Café
Excerpt: Understanding n-transport in terms of Schreier theory for groupoids.
Tracked: September 5, 2006 3:25 PM

Re: 10D SuGra 2-Connection

Further to the comments on inner automorphisms of a 2-group above:

Stefano Kasangian, Giuseppe Metere and Enrico M. Vitale

Split extensions, semidirect product and holomorph of categorical groups Homology Homotopy Appl. 8, no. 1 (2006),

Abstract: Working in the context of categorical groups, we show that the semidirect product provides a biequivalence between actions and points. From this biequivalence, we deduce a two-dimensional classification of split extensions of categorical groups, as well as the universal property of the holomorph of a categorical group. We also discuss the link between the holomorph and inner autoequivalences

If nothing else, it goes into the depths of

$G_2 \rtimes \mathcal{E}q(G_2)$

for $G_2$ a 2-group.

Posted by: David Roberts on September 6, 2006 4:49 AM | Permalink | Reply to this
Read the post Puzzle Pieces falling into Place
Weblog: The n-Category Café
Excerpt: On the 3-group which should be underlying Chern-Simons theory.
Tracked: September 28, 2006 3:35 PM
Read the post A 3-Category of twisted Bimodules
Weblog: The n-Category Café
Excerpt: A 3-category of twisted bimodules.
Tracked: November 3, 2006 2:18 PM
Read the post Chern-Simons Lie-3-Algebra inside derivations of String Lie-2-Algebra
Weblog: The n-Category Café
Excerpt: The Chern-Simons Lie 3-algebra sits inside that of inner derivations of the string Lie 2-algebra.
Tracked: November 7, 2006 8:56 PM
Read the post Nicolai on E10 and Supergravity
Weblog: The n-Category Café
Excerpt: H. Nicolai on further progress in checking the hypothesis that the dynamics of supergravity is encoded in geodesic motion on a Kac-Moody group coset.
Tracked: November 29, 2006 6:31 PM
Read the post Zoo of Lie n-Algebras
Weblog: The n-Category Café
Excerpt: A menagerie of examples of Lie n-algebras and of connections taking values in these, including the String 2-connection and the Chern-Simons 3-connection.
Tracked: May 10, 2007 6:20 PM
Read the post String and Chern-Simons Lie 3-Algebras
Weblog: The n-Category Café
Excerpt: A talk on Chern-Simons Lie n-algebras.
Tracked: August 10, 2007 6:17 PM

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