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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)G_{(2)}


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

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

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 nn-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

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 nn-Transport (pdf slides, blog)

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

Hisham Sati, U.S., Jim Stasheff
L L_\infty-Connections and Applications to String- and Chern-Simons nn-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 nn-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 nn-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

I started to describe this step classicalChernSimons3transportquantizationquantumChernSimons3transport 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<3d \lt 3-dimensional manifolds. For its value on (d=3)(d=3)-dimensional manifolds (i.e. for the “path integral”) one will need BV-quantization.

How BV quantization arises in the context of nn-transport is indicated in the last section of L 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 GG 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 BG\mathbf{B} G transgressed to the groupoid Funct(B,BG)Funct(\mathbf{B}\mathbb{Z}, \mathbf{B} G).

I think this is a very good point of view. It’s precisely the statement that nn-transport is quantized to quantum nn-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 nn-groups or Lie nn-algebras.

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

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:classicalntransportquantomquantize : classical n-transport \to quantom ntransport-transport, namely understanding what it does on d<nd \lt n-morphisms.

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

a) transport nn-functors on cobordism nn-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 nn-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 nn-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 nn-transport – I am thinking that the existing presciptions for integrating Lie algebroids (Weinstein, et al) and L L_\infty-algebra (Getzler, Henriques) are best thought of as forming the fundamental \infty-groupoids BG=Bexp(g):=Π (X CE(g)) \mathbf{B} G = \mathbf{B} exp(g) := \Pi_\infty(X_{CE(g)}) of generalized smooth spaces X CE(g)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 nn-transport: the \infty-category of smooth \infty-functors from \infty-paths in a smooth space XX to smooth \infty-groups G=exp(g)G = exp(g) is equivalent to the corresoponding \infty-groupoid of gg-valued differential forms (gg an L L_\infty-algebroid) on XX.

It feels like this should collaps to a big general abstract nonsense tautology after realizing that we are talking about smooth morphisms Π (X)Π (X CE(g)). \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
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