The n-Category Café

Thanks a lot for your comment. I will reply to it as I understand it. In case you should feel that my replies are missing your point or are being otherwise unsatisfactory, please be so kind and say so.

give a categorical understanding of differential (de Rham) cohomology.

I’d say for this case what I am trying to do is already understood, namely

an element of ${\overline{H}}^2(X)$ is a line bundle on X, with connection.

In terms of the language that I am trying to express things in, this would read:

an element of ${\overline{H}}^2(X)$ is an equivalence class in the category of smooth functors from $P_1(X)$ to $1D\mathrm{Vect}$.

(Here $P_1(X)$ denotes some notion of paths in $X$ and I’d have to tell you what I mean by that functor being smooth. It’s what you would expect anyway. I plan to talk about the details in an upcoming entry.)

A similar statement applies to $H^3(X)$. Here $P_1(X)$ is replaced by $P_2(X)$ and 1-D vector spaces by something like 1-dimensional 2-vector spaces.

“And so on.”

In other words, ${\overline{H}}^n(X)$ classifies abelian $(n-2)$-gerbes with connection. Since everything is abelian, all the subtleties that I tried to express in the above entry disappear.

I think what I am trying to understand in functorial language are exactly phenomena like that “shifted quantization condition”. Namely, I would like to understand 2-gerbes with connection, whose connection 3-form is - for instance - while abelian, locally built from Chern-Simons 3-forms of some non-abelian bundle (=0-gerbe).

The idea that this is to be understood in terms of a 2-gerbe with a gauge 3-group which is expressible in terms of three ordinary groups, one of which is abelian, while the others are not. This would be a non-abelian 2-gerbe and it turns out that in order to understand connections on these one has to face the issue of “fake curvatures” that I addressed above.

I have given some technical details on the Lie 3-algebra of such a Chern-Simons 2-gerbe in the entry on the sugra 3-connection ($\to$).

The existence of other generalized differential cohomology theories. The RR fields of Type II string theory live in differential K-theory

Yes, we had some discussion on that somewhare in the comment sections.

Right now I have no good idea about how to conceive anything involving RR fields directly using functorial language.

All I can offer at the moment is the observation, that up in 11-dimensions, the RR-fields should reassamble with the KR 2-form to a single 2-gerbe with 3-form connection.

So I think in 11D we need to understand nonabelian 2-gerbes. We should try to understand what it means to KK-reduce such a 2-gerbe. The result should be a 1-gerbe plus other stuff - the RR stuff. I have as yet no good idea at all if that other stuff has a useful functorial description by itself.

Where is local supersymmetry?

In the structure 3-group.

That was, for me, the big insight of translating the FDA-sugra description into categorical terms. This shows that - classically and locally - 11D sugra is the theory of a 3-connection with values in a Lie 3-algebra which is cooked up from the super-Poincaré group in degree 1, has nothing in degree 2 and looks like $U(1)$ in degree 3.

Let this 3-algebra be denoted by $\mathrm{SPoin}(11)$. Let the 3-algebra of an $E_8$-Chern-Simons 2-gerbe be denoted by $\mathrm{CS}{E}_8$.

Then we may try to study 2-gerbes with connection taking values in $\mathrm{SPoin}(11)\oplus \mathrm{CS}{E}_8$.

Locally, such a connection encodes precisely the field content of 11D sugra - including the gravitino - together with a connection on an $E_8$-bundle. The 3-form part of the connection is automatically built from a $\mathrm{Lie}(U(1))$-valued part and the Chern-Simons 3-form of that $E_8$-bundle.

What I don’t understand yet is how to force the contrtribution from the Poincaré CS-3-form into the game.

On the other hand, what the discussion in the above entry is supposed to solve is in which sense precisely we have to conceive that “3-connection” in order that imposing a certain flatness condition on it (a Bianchi identity, really) implies the (classical) equations of motion of supergravity.

I could say more, but maybe I should stop here for the moment. I have to run anyway.