The n-Category Café

Okay, I guess my problem was that I know what you meant where your slides say:

There is also a Lie 3-supergroup 2-Brane(n; 1) associated with super-2-branes.

But of course anyone who hangs out here should know that a 3-group can be defined as a kind of tricategory.

Thanks! What’s the simplest way to see what makes $\gamma = -1/32$ special?

What’s the simplest way to see what makes $\gamma = - \frac{1}{32}$ special?

As you know, these prefactors are generally a tremendous source of headaches, that come in two stages: first those related to the sorting out of all conventions (your own and those of all the references) and second those coming from the actual computation.

I don’t think I can give an easy way to see why it is that number and not another. It is fixed by the given fermion anomaly term in 10 D sugra and the way it has to be canceled in the GS mechanism. At the moment I am just trusting Bonora et al. that they can count correctly.

The main message for the moment is that both the bosonic and the fermionic cocycle need to be added together, in a uniquely fixed way. (Where “need to” is in the sense of: such that the corresponding super-Lie 2-group is precisely such that lifts of differential cocycles from $SISO$ to it correspond exactly to the obstructions/anomalies seen in string physics).

I’ve seen that article, but John Huerta has read it more carefully than I have.

Part of the difference is that they use the computer algebra system LiE to compute the cohomology of superPoincaré Lie algebras for spacetimes of dimension 6 and 10, while in SUSY2 we used normed division algebras to give a unified proof that certain 3-cocycles in dimensions 3, 4, 6, and 10 and 4-cocycles in dimensions 4, 5, 7 and 11 are nontrivial.

Of course, I don’t claim that this is a complete account of what they do!

Right. Movshev et al. use the LiE software to decompose the cochain groups of the supertranslation algebras into irreps of $\mathfrak{so}(n)$, for $n=6$ and $n=10$. They use this information to prove the cohomology rings are as stated on p. 4 and p. 6.

The relevant piece of cohomology is, in both cases:

$H^{4,1}$

This contains an $\mathfrak{so}(n)$-invariant piece. This is generated by the fermionic Chern-Simons term you mention above, and it is the only part of the supertranslation algebra cohomology to contribute to $\mathfrak{siso}(n)$’s cohomology.

Hi, Urs. I’m reading!

I do not have a nice abstract definition of integration, just this funny procedure that works for the special case that I consider. The special case is this:

Integrate a slim, nilpotent Lie $n$-superalgebra to a slim, nilpotent Lie $n$-supergroup.

A slim, nilpotent Lie $n$-superalgebra is an $L_\infty$-superalgebra with zero differential that is trivial except for degrees 0 and $n-1$, and which is nilpotent in the usual sense (nested brackets are eventually zero).

This reduces to the following data: a nilpotent Lie superalgebra $\mathfrak{n}$ with a representation $\mathfrak{h}$ and an $\mathfrak{h}$-valued Lie superalgebra $(n+1)$-cocycle on $\mathfrak{n}$.

A slim, Lie $n$-supergroup is a skeletal, globular $(n+1)$-groupoid internal to supermanifolds, with the one-point supermanifold as objects, trivial morphisms except at level 1 and level $n$, and no coherence ismorphisms besides the associatorator…ator. It is nilpotent in the sense that the Lie supergroups of 1-morphisms and $n$-morphisms are both nilpotent.

This reduces to the following data: A nilpotent Lie supergroup $N$ which acts by automorphism on an abelian Lie supergroup $H$, and an $H$-valued Lie supergroup $(n+1)$-cocycle on $N$.

I like Sachse’s article about supergeometry a lot. It’s certainly the clearest one I have read, though I have been thinking about the same kind of thing a lot to work on my thesis, and that helps to make it clear. The equivalence between superalgebra over $\mathbb{R}$ and ordinary algebra over what Sachse calls $\overline{\mathbb{R}}$ is a cornerstone of my thesis which I have never seen quantified so clearly elsewhere.

Now, a question for you. Let $G$ be a Lie supergroup with Lie superalgebra $\mathfrak{g}$. You talk of applying the general procedure for integrating Lie $\infty$-algebras to $\mathfrak{g}$. Let’s call the $\infty$-supergroup we get out of this $\mathcal{G}$.

You say:

For instance the formal Lie integration of a super Lie algebra in this context yield in general not a delooped group object, but a genuine groupoid object: in degree 0 it has not the point, but a superpoint.

To me, this means that $\mathcal{G}$ differs from the shifted supergroup $\mathbf{B}G$ in that $\mathbf{B}G$ has a single point as its objects, while $\mathcal{G}$ has some superpoint as objects. But if I hand you a Lie superalgebra $\mathfrak{g}$, can you tell me which superpoint I’ll get as the objects of $\mathcal{G}$?

But if I hand you a Lie superalgebra $\mathfrak{g}$, can you tell me […]

Right, let me spell that out. In fact I believe what I said above was misled, and that what I say now will harmonize with what you have:

First briefly recall the story of Lie integration for ordinary $L_\infty$-algebras: given an $L_\infty$-algebra $\mathfrak{g}$ with Chevalley-Eilenberg algebra $CE(\mathfrak{g})$, the bare $\infty$-groupoid (not remembering mooth structure) that it universally integrates to (“universally” in the sense of “universal cover”, the actual $n$-groupoids that we are interested in are truncations of this) is the Kan complex

$$\exp(\mathfrak{g}) : [k] \mapsto \{\Omega^\bullet(\Delta^k) \leftarrow CE(\mathfrak{g})\} \,.$$

This becomes a smooth $\infty$-groupoid by remembering what the $U$-parameterized smooth families of $k$-morphisms are, and these we take to be

$$\exp(\mathfrak{g}) : (U, [k]) \mapsto \{\Omega^\bullet_{vert}(U \times \Delta^n) \leftarrow CE(\mathfrak{g})\} \,,$$

where on the left now we have the complex of differential forms on $U \times \Delta^n$ that 1. have sitting instants towards the boundaries of the simplex and 2. are vertical with respect to the projection $U \times \Delta^n \to U$.

(For all details see the above link.)

Now we generalize this to super-$L_\infty$-algebras $\mathfrak{g}$. First consider again the bare case, without smooth, but with superstructure.

Then by Molotkov-Schwarz and others we are to produce a simplicial presheaf on super points to get the bare super $\infty$-groupoid

$$\exp(\mathfrak{g}) : (\mathbb{R}^{0|q}, [k]) \mapsto \{\Omega^\bullet_{vert}(\mathbb{R}^{0|q} \times \Delta^n) \leftarrow CE(\mathfrak{g}) \} \,,$$

whre again the “vertical” is with respect to the evident projection. So if we write $\Lambda_q := C^\infty(\mathbb{R}^{0|q})$ for the Grassmann algebra of functions on this superpoint, this is

$$\exp(\mathfrak{g}) : (\mathbb{R}^{0|q}, [k]) \mapsto \{\Lambda_q \otimes_{\mathbb{R}}\Omega^\bullet(\Delta^n) \leftarrow CE(\mathfrak{g}) \} \,.$$

Finally, to make this super simplicial set a smooth super simplicial set, hence a smoth super $\infty$-groupoid we add again the probes by smooth test spaces $U = \mathbb{R}^n$ to get in total

$$\exp(\mathfrak{g}) : (\mathbb{R}^n , \mathbb{R}^{0|q}, [k]) \mapsto \{\Omega^\bullet_{vert}(\mathbb{R}^{n|q} \times \Delta^n) \leftarrow CE(\mathfrak{g}) \} \,,$$

where the verticality of the forms is with respect to the full projection $\mathbb{R}^{n|q} \times \Delta^k \to \mathbb{R}^{p|q}$.

This way, we indeed get that this thing is connected, hence is of the form

$$\cdots = \mathbf{B} G$$

for $G$ a smooth super $\infty$-group integrating $\mathfrak{g}$.

To give an example: for $\mathfrak{g}$ the super Poincaré Lie algebra for which $CE(\mathfrak{g})$ is generated from $(e^a)$ and $(\omega^{a b})$ in degree $(1,even)$ and from $(\psi)$ in degree $(1,odd)$ with the differential defined by

$$d \omega^{a b} = \omega^{a}{}_c \wedge \omega^{c b}$$

$$d e^a = \omega^{a}{}_b \wedge e^b + \bar \psi \Gamma^a \wedge \psi$$

$$d \psi = \frac{1}{4} \omega^{a b}\Gamma_{a b} \wedge \psi$$

we have over the trivial superpoint $q = 0$ that $\exp(\mathfrak{iso})$ is the ordinary universal Lie integration of the Poisson Lie algebra (all $\psi$s always have to map to 0-forms), whereas over the odd line we find in

$$\exp(\mathfrak{iso})_{\mathbb{R}^{n|1}, [1]}$$

also contributions $\Omega^\bullet_{vert}(\mathbb{R}^{n|1} \times \Delta^1 )$ of the form

$$\psi \mapsto \theta f d t \,,$$

where $f \in C^\infty(\mathbb{R}^n)$, where $\theta$ is the canonical coordinate on $\mathbb{R}^{0|1}$ and $t$ that on $\Delta^1$.

In other words, for fixed $q$ the remaining simplicial presheaf on $\mathbb{R}^n$s is that which integrates the ordinary (bosonic) Lie algebra $(\mathfrak{g} \otimes \Lambda_q)_{even}$.

(All this should be obvious, I am just saying it so that we all know what I mean. )