## February 6, 2008

### 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 $Lie(G_{(2)})$:

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

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.

Posted at February 6, 2008 10:00 AM UTC

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### Re: Smooth 2-Functors and Differential Forms

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

Posted by: Tim Silverman on February 7, 2008 12:01 AM | Permalink | Reply to this

### Re: Smooth 2-Functors and Differential Forms

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

I guess it is.

There are a bunch of further puzzle pieces which I think I have an idea how to fill in, but which need more time to work out in careful detail.

Let me use this opportunity to list a few, in case anyone feels like looking into them:

- The Lie version of Willerton’s interpretation of Freed-Hopkins-Teleman. Simon Willerton explained that a very nice point of view of FHT is obtained in the toy version of finite groups $G$ by (slightly, but not much, rephrased by me here) understanding it as being about the space of sections of the Chern-Simons 3-bundle over the groupoid $\mathbf{B} G$ transgressed to the groupoid $Funct(\mathbf{B}\mathbb{Z}, \mathbf{B} G)$.

I think this is a very good point of view. It’s precisely the statement that $n$-transport is quantized to quantum $n$-transport by transgressing to cobordisms and taking sections there.

And I think it does generalize to the full Lie version involving Kac-Moody loop groups proper.

There are two ways to do this: using Lie $n$-groups or Lie $n$-algebras.

Using Lie $n$-groups, I discussed in The 2-monoid of observables on $String(G)$ (pdf, blog) how $\mathrm{Funct}(\mathbf{B}\mathbb{Z}, \mathbf{B} String_G)$ yields the groupoid whose reps are twisted equivariant vector bundles on $G$.

The corresponding Lie $\infty$-algebraic description I began describing in loop Lie algebroids (pdf (section 2.3), blog).

The induced description of the fusion product I indicated in Fusion and String Field Star Product (blog).

All this is directly relevant for tackling Chern-Simons and WZW theory. But the underlying mechanism is much more general. This is really all about understanding the kinematics of the quantization functor $quantize : classical n-transport \to quantom$n$-transport$, namely understanding what it does on $d \lt n$-morphisms.

-the relation between $n$-functorial QFT and AQFT There exist two different axiom systems for defining what quantum field theory actually is:

a) transport $n$-functors on cobordism $n$-categories

b) cosheaves of local algebras.

I think both points of viewes are two sides of one single thing and it would be very useful to relate both approaches more closely

In Local Nets from 2-Transport (pdf, blog) I describe what I think is going on: AQFT is the image of quantum $n$-transport under postcomposition with a functor that forms endomorphism algebras:

I wish I’d find more time pushing this further, since there is a wealth of insights bound to be hidden here. I made some remarks about various relations here and at other places.

- deriving the BV-master equation and the perturbative BV path integral from $n$-transport – there are indications that BV-formalism is secretly the Lie $\infty$-algebra version of a theory of canonical measures (I, II, III) (“integrals without integrals”) on $\infty$-groupoids. Various aspects of BRST-BV quantization I think I can already interpret and derive that way (I, II, III,IV), but I am still not sure about the path integral itself (but that’s not too surprising…)

- integration of Lie $\infty$-algebras using fundamental $\infty$-groupoids of generalized smooth spaces and the tautologification of smooth $n$-transport – I am thinking that the existing presciptions for integrating Lie algebroids (Weinstein, et al) and $L_\infty$-algebra (Getzler, Henriques) are best thought of as forming the fundamental $\infty$-groupoids $\mathbf{B} G = \mathbf{B} exp(g) := \Pi_\infty(X_{CE(g)})$ of generalized smooth spaces $X_{CE(g)}$. I describe this in Differential forms and smooth spaces.

To the extent that this is right and works, it would be just a repackaging of what’s already being done. But it should be helpful for proving the

obvious central conjecture about $n$-transport: the $\infty$-category of smooth $\infty$-functors from $\infty$-paths in a smooth space $X$ to smooth $\infty$-groups $G = exp(g)$ is equivalent to the corresoponding $\infty$-groupoid of $g$-valued differential forms ($g$ an $L_\infty$-algebroid) on $X$.

It feels like this should collaps to a big general abstract nonsense tautology after realizing that we are talking about smooth morphisms $\Pi_\infty(X) \to \Pi_\infty(X_{CE(g)}) \,.$ But I am really not sure yet about this.

But if we could do this, it should help to take all of Lie $\infty$-connections and integrate it up systematically.

Posted by: Urs Schreiber on February 7, 2008 10:12 PM | Permalink | Reply to this
Read the post Construction of Cocycles for Chern-Simons 3-Bundles
Weblog: The n-Category Café
Excerpt: On how to interpret the geometric construction by Brylinksi and McLaughlin of Cech cocycles classified by Pontrjagin classes as obstructions to lifts of G-bundles to String(G)-2-bundles.
Tracked: February 12, 2008 1:39 PM
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Weblog: The n-Category Café
Excerpt: On the notion of nonabelian differential cohomology.
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Weblog: The n-Category Café
Excerpt: A discussion of differential nonabelian cocycles classifying higher bundles with connection in the context of the general theory of descent and cohomology with coefficients in infnity-category valued presheaves as formalized by Ross Street.
Tracked: March 22, 2008 7:50 PM
Read the post Connections on Nonabelian Gerbes and their Holonomy
Weblog: The n-Category Café
Excerpt: An article on transport 2-functors.
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Weblog: The n-Category Café
Excerpt: A talk on parallel 2-transport.
Tracked: September 6, 2008 4:38 PM
Read the post Twisted Differential Nonabelian Cohomology
Weblog: The n-Category Café
Excerpt: Work on theory and applications of twisted nonabelian differential cohomology.
Tracked: October 30, 2008 7:47 PM

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