## August 21, 2006

### n-Curvature

#### Posted by Urs Schreiber

The concept of $n$-curvature of $n$-transport - and the nature of “fake” curvature.

An abstract definition of $n$-curvature, suitable for use with the notion of $n$-transport ($\to$) is given here

$\;\;\;$ Curvature.

Apart from the definitions, this text contains just the baby example of the curvature of a Lie group valued 1-transport, reproducing the notion of a Lie-algebra valued 1-form $A$, its curvature 2-form $F_A = dA + A \wedge A$ and the corresponding Bianchi identity $d_A F_A = 0$.

The first nontrivial example, that played a great role in motivating these abstract definitions, is that of curvature of a 2-transport with values in a strict Lie 2-group. The general concept of $n$-curvature, as described in the above pdf, explains why the $G_2$-valued 2-transport described before ($\to$) is just a special case of what one would more generally want to understand under principal 2-transport with values in a Lie 2-group.

This is explained and worked out here:

$\;\;\;$ $\Sigma(\mathrm{Inn}(G_2))$-2-Transport.

In particular, the nature of nonvanishing “fake curvature” in the context of 2-transport is clarified by this.

A quick way to derive these results at the differential level is to use FDA techniques ($\to$). Those who know how to use these to describe morphisms of Lie $n$-algebras may find the FDA-version of the above $\Sigma(\mathrm{Inn}(G_2))$-2-transport at the end of this file:

$\;\;\;$ FDA Laboratory.

Posted at August 21, 2006 11:21 PM UTC

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### Re: n-Curvature

08 19 06

Hello guys:
This is an awesome blog. I have been stepping around your String coffee table site for a while now, but don’t have all of the mathematical background to be a reasonable commenter. So for now, I am learning and soaking up information. I will advert this site on my blog because it stands to be an EXCELLENT resource. I know it will take some time to get it off of the ground, but look forward to its time evolution. In the meantime, your current post demands my attention:)
Warm Regards:)

Posted by: Mahndisa on August 19, 2006 2:57 PM | Permalink | Reply to this

### Re: n-Curvature

Since Mahndisa posted a general comment, maybe I should jump in with a general comment.

It seems to me that what you are doing here is attempting to give a categorical understanding of differential (de Rham) cohomology. There’s a well-known model for the second differential cohomology group, namely an element of $\check{H}^2(X)$ is a line bundle on $X$, with connection. Up to the (important!) matter of the shifted quantization condition for the field strength, the 3-form gauge field of 11-D supergravity is an element of $\check{H}^4(X)$.

It’s that latter group that you seem to be trying to find a nice description for.

But there are three things that strike me as missing:

1. The aforementioned shifted quantization condition (which is very much not a statement about the classical theory).
2. The existence of other generalized differential cohomology theories. The RR fields of Type II string theory live in differential K-theory.
3. Where is local supersymmetry? In your previous entry, you cite Castellani for the statement that the other fields of 11D supergravity can also be fit into this picture. But, unless I completely miss the point, everything you say looks very bosonic. Where’s the gravitino?
Posted by: Jacques Distler on August 19, 2006 3:58 PM | Permalink | PGP Sig | Reply to this

### Re: n-Curvature

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 $\bar H^2(X)$ 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 $\bar H^2(X)$ 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, $\bar H^n(X)$ 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.

Posted by: urs on August 20, 2006 8:06 PM | Permalink | Reply to this
Read the post On n-Transport, Part II
Weblog: The n-Category Café
Excerpt: Smooth transport, differentials of transport, and nonabelian differential cocycles.
Tracked: August 21, 2006 8:41 PM
Read the post Picturing Morphisms of 3-Functors
Weblog: The n-Category Café
Excerpt: Diagrams governing morphisms of 3-functors.
Tracked: August 25, 2006 6:14 PM
Read the post 10D SuGra 2-Connection
Weblog: The n-Category Café
Excerpt: On the Lie 2-algebra governing 10-dimensional supergravity.
Tracked: August 28, 2006 3:50 PM

### Update, 31. Aug. 06

I have been busy computing transitions of smooth 2-functors from 2-paths to the 3-group $\mathrm{Inn}(G_2)$, for $G_2$ a strict 2-group.

I am now done with a first run through all the computations. What I find is this:

A transition tetrahedron for $\Sigma(\mathrm{Inn}(G_2))$-2-transport does indeed imply the nonabelian differential cocycles found by Breen&Messing and Aschieri&Jurčo from nonabelian gerbes with arbitrary, not necessarily fake flat, connection.

One noteworthy result is this:

I find that the $\Sigma(\mathrm{Inn}(G_2))$-transitions are slightly more general than what these authors have. I find four more kinds of differential forms involved in the transitions than these authors have. The cocycles found by these authors follow by setting these forms to zero.

Maybe I made a mistake. Currently I don’t think so. Instead, I expect that this is due to fact that I am really looking at flat 3-transport. So I think we see gauge trivial cocycle data of 2-gerbes appearing.

Another point, where it is more likely that I made a mistake, is this: there is a transition 2-form $d_{ij}$ on double overlaps for the non-fake flat case. I find this 2-form as well as the correct transition law as given in Aschieri&Jurčo (I don’t see this law in Breen&Messing stated).

Almost what they have, that is. My law is lacking precisely one of the terms that Aschieri&Jurčo have.

The diagram this law comes from in my formalism is a 3-commuting thing, hence rather involved. So it could be that I simply missed a term that should be there.

But since everything else matches, it is clear that the general thing is right.

The new calculations are all in here.

Posted by: urs on August 31, 2006 1:35 PM | Permalink | Reply to this
Read the post n-Transport and Higher Schreier Theory
Weblog: The n-Category Café
Excerpt: Understanding n-transport in terms of Schreier theory for groupoids.
Tracked: September 5, 2006 3:24 PM
Read the post On n-Transport: 2-Vector Transport and Line Bundle Gerbes
Weblog: The n-Category Café
Excerpt: Associated 2-transport, 2-representations and bundle gerbes with connection.
Tracked: September 7, 2006 2:00 PM
Weblog: The n-Category Café
Excerpt: A quest for arrow-theoretic differential geometry.
Tracked: September 20, 2006 9:15 PM
Read the post Bulk Fields and induced Bimodules
Weblog: The n-Category Café
Excerpt: Bulk field insertions in 2D CFT in terms of 2-transport: endomorphisms of 2-monoids.
Tracked: September 27, 2006 5:26 PM
Read the post Puzzle Pieces falling into Place
Weblog: The n-Category Café
Excerpt: On the 3-group which should be underlying Chern-Simons theory.
Tracked: September 28, 2006 3:33 PM
Read the post WZW as Transition 1-Gerbe of Chern-Simons 2-Gerbe
Weblog: The n-Category Café
Excerpt: How the WZW 1-gerbe arises as the transition 1-gerbe of the Chern-Simons 2-gerbe.
Tracked: October 29, 2006 5:10 PM
Read the post A 3-Category of twisted Bimodules
Weblog: The n-Category Café
Excerpt: A 3-category of twisted bimodules.
Tracked: November 3, 2006 2:15 PM
Read the post Chern-Simons Lie-3-Algebra inside derivations of String Lie-2-Algebra
Weblog: The n-Category Café
Excerpt: The Chern-Simons Lie 3-algebra sits inside that of inner derivations of the string Lie 2-algebra.
Tracked: November 7, 2006 8:55 PM
Read the post Local Transition of Transport, Anafunctors and Descent of n-Functors
Weblog: The n-Category Café
Excerpt: Conceps and examples of what would be called transition data or descent data for n-functors.
Tracked: December 8, 2006 9:06 AM
Read the post QFT of Charged n-Particle: The Canonical 1-Particle
Weblog: The n-Category Café
Excerpt: On the category of paths whose canonical Leinster measure reproduces the path integral measure appearing in the quantization of the charged particle.
Tracked: March 19, 2007 9:02 PM
Read the post Oberwolfach CFT, Tuesday Morning
Weblog: The n-Category Café
Excerpt: On Q-systems, on the Drinfeld Double and its modular tensor representation category, and on John Roberts ideas on nonabelian cohomology and QFT.
Tracked: April 3, 2007 2:07 PM
Weblog: The n-Category Café
Excerpt: The n-curvature canonically associated with a transport n-functor.
Tracked: April 25, 2007 7:53 PM
Read the post The First Edge of the Cube
Weblog: The n-Category Café
Excerpt: The notion of smooth local i-trivialization of transport n-functors for n=1.
Tracked: May 4, 2007 9:00 PM
Read the post Zoo of Lie n-Algebras
Weblog: The n-Category Café
Excerpt: A menagerie of examples of Lie n-algebras and of connections taking values in these, including the String 2-connection and the Chern-Simons 3-connection.
Tracked: May 10, 2007 6:07 PM
Read the post Curvature, the Atiyah Sequence and Inner Automorphisms
Weblog: The n-Category Café
Excerpt: On the notion of curvature 2-functor in light of morphisms from the path sequence of the base to the Atiyah sequence of the bundle.
Tracked: June 20, 2007 5:04 PM
Read the post The inner automorphism 3-group of a strict 2-group
Weblog: The n-Category Café
Excerpt: On the definition and construction of the inner automorphism 3-group of any strict 2-group, and how it plays the role of the universal 2-bundle.
Tracked: July 4, 2007 12:56 PM
Read the post Arrow-Theoretic Differential Theory
Weblog: The n-Category Café
Excerpt: We propose and study a notion of a tangent (n+1)-bundle to an arbitrary n-category. Despite its simplicity, this notion turns out to be useful, as we shall indicate.
Tracked: July 30, 2007 4:00 PM
Read the post n-Curvature, Part III
Weblog: The n-Category Café
Excerpt: Curvature is the obstruction to flatness. Believe it or not.
Tracked: October 16, 2007 10:51 PM
Read the post What has happened so far
Weblog: The n-Category Café
Excerpt: A review of one of the main topics discussed at the Cafe: Sigma-models as the pull-push quantization of nonabelian differential cocycles.
Tracked: March 27, 2008 4:49 PM
Read the post Questions on n-Curvature
Weblog: The n-Category Café
Excerpt: Some questions for Urs.
Tracked: September 26, 2009 5:46 PM

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