### Thoughts (Mostly on Super ∞-Things)

#### Posted by Urs Schreiber

I am on leave of absence from Hamburg and spending some time travelling before the program in Bonn starts next month. After a very productive week with Jens Fjelstad in Denmark this is now my last evening at Notre Dame, where I spent a very pleasant time with Stephan Stolz.

Time went by quickly, filled with discussions, mostly on nonabelian differential cohomology # and on superQFT, and many thoughts want to be further developed now.

I had planned to collect notes on some such thoughts last Sunday, but Jim Stasheff and Hisham Sati rightly pushed me to work on finalizing our article on Fivebrane structures (section 3), following up the one on $L_\infty$-connections #.

One claim is that our $L_\infty$-algebraic connection descent objects (section 7) may be integrated by hitting them with the functor $DGCA \stackrel{form classifying space}{\to} SmoothSpace \stackrel{form path \omega-groupoid}{\to} Smooth\omega-Cat$ to yield nonabelian differential cocycles #, a process reproducing the construction of cocycles by Brylinski-McLaughlin #.

(Fun exercise: read their article and identify how they are secretly integrating $L_\infty$-algebras to $\omega$-groups and $L_\infty$-connections to differential cocyles.)

But that must wait now until later. With a little luck Hisham will be around at UPenn this week, where I’ll go tomorrow to visit Jim, and we’ll see further.

But Lie $n$-tegration has many aspects. Below some comments on super parallel $n$-transport (see Florin Dumitrescu’s thesis for a nice discussion of the $n=1$ case) and integration of *super*-$L_\infty$-algebras (such as our favorite one (page 54)) to smooth *super* $\omega$-groups.

What’s a super $L_\infty$-algebra? (see here for what an ordinary $L_\infty$-algebra is.) Let $SVect$ be the category of $\mathbb{Z}_2$-graded vector spaces equipped with the unique non-trivial symmetric braiding. A super $L_\infty$-algebra is an $L_\infty$-algebra internal to that. So, dually, it’s a $\mathbb{N}$-$\mathbb{Z}_2$-bigraded quasi-free differential graded commutative algebra.

The archetypical example of such a beast is the super tangent Lie algebroid of any smooth superspace $X$: the dual dgc algebra is just that of super-differential forms $\Omega^\bullet(X)$ on $X$. For me here, a smooth superspace is a sheaf on the site $S$ whose objects are superEuclidean spaces and whose morphisms are smooth maps between these $Obj(S) = \mathbb{N}\times\mathbb{N}$ $S(n|n',m|m') = SmoothSupermanifolds(\mathbb{R}^{n|n'},\mathbb{R}^{m|m'}) \,.$

Given any super $L_\infty$-algebra $g$ we get the smooth superspace $S(CE(g))$ classifying $g$-valued super-forms using the usual adjunction coming from the ambimorphic object $\Omega^\bullet$ (you are hearing my teacher Todd Trimble speak through me, here) $S(CE(G)) : \mathbb{R}^{n|n'} \mapsto Hom_{superDGCA}(CE(g),\Omega^\bullet(\mathbb{R}^{n|n'})) \,.$ This classifies (section 6.5) $g$-valued forms in that for any smooth super $X$ $SmoothSuperspaces(X,S(CE(g))) \simeq \Omega^\bullet(X,g) \,.$

A thin homotopy class of a superpath in that classifying space $\gamma : \mathbb{R}^{1|1} \to S(CE(g))$ is an element in the super $\omega$-group integrating $g$. Continuing this way, we form the super path $\omega$-groupoid

$\mathbf{B}G := \Pi_\omega^N(S(CE(g)))$

of the classifying space, for any $N \in \mathbb{N}$, whose $k$-morphisms are thin homotopy classes of $(k|N)$-dimensional paths in the space $\Sigma : \mathbb{R}^{k|N} \to S(CE(g))$ (section 2.3)

Voila, the $\omega$-supergroup $G$ integrating $g$.

Now take any (flat for brevity) $g$-valued superform on $X$, represented by a superDGCA morphism $\Omega^\bullet(Y) \stackrel{A}{\leftarrow} CE(g)$ and hit that entire morphism with our integration functor $\Pi_\omega^N \circ S : superDGCA \to SuperSmooth\omega Cat$ to produce its corresponding super-parallel transport $tra_A : \Pi_\omega^N(Y) \to \mathbf{B}G \,.$ And so on.

Here is a somewhat more philosophical remark. (Experience shows that it’s of the kind that I tend to find deeply puzzling while you are likely to find it at best pointless.)

I keep thinking about what superification *really* means. For instance, I keep noticing that superification shares many properties with categorification: categorification introduces $\mathbb{N}$-gradings, while superification introduces $\mathbb{Z}_2$-gradings.

I have that funny trait that I feel annoyed by the fact that $\mathbb{R}^{n|n'}$ is defined using graded algebras, somewhat messing with the Zen-like beauty of space and quantity, only for $\Omega^\bullet(\mathbb{R}^{n|n'})$ later to introduce yet another layer of graded algebra.

So I observe this: little to nothing of the above changes if we do a slight repackaging of concepts:

instead of regarding the classifying space $S(CE(g))$ of a super $L_\infty$-algebra $g$ as a smooth superspace, I regard it as an ordinary smooth space probed by ordinary Euclidean spaces by setting $S(CE(g)) : \mathbb{R}^n \mapsto Hom_{superDGCA}(CE(g),\Omega_N^\bullet(\mathbb{R})) \,,$ where now $\Omega^\bullet_N : Euclidean domains \to superDGCAs$ sends $\mathbb{R}^n$ to the superDGCA of superdifferential forms on $\mathbb{R}^{n|N}$.

The point being that odd superfunctions are so close in concept to differential forms that we can just repackage them with the forms and assign the full package to any ordinary space.

These are lots of words for the simple fact that using the hom-adjunction we can always think $\begin{aligned} \Omega^\bullet(\mathbb{R}^{n|N}) &:= SuperSmoothSpaces(\mathbb{R}^{n|N},\Omega^\bullet(-)) \\ &= SuperSmoothSpaces(\mathbb{R}^{n|0}\times \mathbb{R}^{0|N},\Omega^\bullet(-)) \\ & \simeq SuperSmoothSpaces(\mathbb{R}^n,hom(\mathbb{R}^{0|N},\Omega^\bullet(-))) \\ & =: SuperSmoothSpaces(\mathbb{R}^n,\Omega^\bullet_N(-)) \\ & =: \Omega^\bullet_N(\mathbb{R}^n) \end{aligned} \,.$ But still. It makes me wonder if we shouldn’t rethink.

## Re: Thoughts (mostly on super infinity-things)

Motivating toy examples and then the case of ordinary Chern-Simons cocycles realizing the first Pontryagin class, obtained from integrating the $L_\infty$-connection that obstructs the lift of a $G$-connection to a $String(G)$ 2-connection, is now in section 5 of my talk notes:

On nonabelian differential cohomology(pdf)