### Smooth 2-Functors and Differential Forms

#### Posted by Urs Schreiber

We have now a detailed description and proof of the relation between

- strict smooth 2-functors on 2-paths in a manifold with values in a strict Lie 2-group $G_{(2)}$

and

- differential forms with values in the corresponding Lie 2-algebra $Lie(G_{(2)})$:

U.S. and Konrad Waldorf
*Smooth Functors vs. Differential Forms*

arXiv:0802.0663v1

Abstract:We establish a relation between smooth 2-functors defined on the path 2-groupoid of a smooth manifold and differential forms on this manifold. This way we set up a dictionary between fundamental notions from category theory and differential geometry. We show that smooth 2-functors appear in several fields, namely as connections on (non-abelian) gerbes, as the curvature of smooth functors and as critical points in BF theory. We demonstrate further that our dictionary provides a powerful tool to discuss the transgression of geometric objects to loop spaces.

This is finally the detailed worked-out version of the stuff I talk about in the entry On $n$-Transport, Part II.

This result is the bridge between the description of ordinary (1-)bundles with connection in terms of parallel transport functors

U.S. and K. Waldorf
*Parallel Transport and Functors*

(arXiv, blog)

and the corresponding description of 2-bundles with connection as indicated in

John Baez, U.S.
*Higher Gauge Theory*

arXiv:math/0511710v2.

The further detailed development of this to a full-blown theory of *2-Transport* will be the content of a followup article which is in preparation. This will sum up various things that Konrad and me have been talking about for a while now, for instance here:

Konrad Waldorf, *Parallel Transport Functors of Principal Bundles and (non-abelian) Bundle Gerbes* (pdf, blog)

Konrad Waldorf, *Parallel Transport and Functors* (pdf slides)

U. S., *Parallel Transport in Low Dimensions* (pdf, blog)

U.S., *On String- and Chern-Simons $n$-Transport* (pdf slides, blog)

The last one also indicates how all this is related to a description where smooth $n$-functors between Lie $n$-groupoids are replaced by morphisms of Lie $n$-algebras/Lie $n$-algebroids. Since the latter are more easily handled for all $n$, on this differential side the entire program has already proceeded past $n=2$ all the way to $\infty$:

Hisham Sati, U.S., Jim Stasheff
*$L_\infty$-Connections and Applications to String- and Chern-Simons $n$-Transport*

(pdf, blog, arXiv).

(The big theorem in the Lie 2-groupoid case in Smooth Functors vs. Differential Forms becomes, in the differential world of Lie 2-algebras, a simple example (see for instance section 6.5.1). Of course then the problem is to integrate that back to the world of Lie $\infty$-groupoids. Dicussion of that can be found here on the blog at On Lie $n$-tegration and Rational Homotopy Theory and Differential Forms and Smooth Spaces.)

As the title suggests, all this is headed towards understanding String- and Chern-Simons $n$-transport (and their higher dimensional generalizations) and their quantization (their image under the “quantization edge” of the cube): in the end we want to hit a classical Chern-Simons parallel 3-transport with the quantization functor to obtain the Chern-Simons quantum 3-transport and the holographically related WZW theory, as described in

Jens Fjelstad and U.S.
*Towards 2-Functorial CFT*

(blog).

I started to describe this step
$classical Chern-Simons 3-transport
\stackrel{quantization}{\to}
quantum Chern-Simons 3-transport$
using the Lie $\infty$-algebraic description in the series of entries *States of Chern-Simons Theory* (I, II, III).

This so far concentrates on just the *kinematics*: the value of the TFT 3-functor on $d \lt 3$-dimensional manifolds. For its value on $(d=3)$-dimensional manifolds (i.e. for the “path integral”) one will need BV-quantization.

How BV quantization arises in the context of $n$-transport is indicated in the last section of $L_\infty$-connections as well as in the notes

U.S., *On the BV-Formalism* (pdf, blog).

But clearly more work is necessary here.

## Re: Smooth 2-Functors and Differential Forms

Wow, it’s really coming together at last, isn’t it?