Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

August 21, 2006

n-Curvature

Posted by Urs Schreiber

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

An abstract definition of nn-curvature, suitable for use with the notion of nn-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 AA, its curvature 2-form F A=dA+AAF_A = dA + A \wedge A and the corresponding Bianchi identity d AF A=0d_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 nn-curvature, as described in the above pdf, explains why the G 2G_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:


\;\;\; Σ(Inn(G 2))\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 nn-algebras may find the FDA-version of the above Σ(Inn(G 2))\Sigma(\mathrm{Inn}(G_2))-2-transport at the end of this file:


\;\;\; FDA Laboratory.


(The discussion about this curvature topic seems to be going on here.)

Posted at August 21, 2006 11:21 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/899

4 Comments & 23 Trackbacks

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 Hˇ 2(X)\check{H}^2(X) is a line bundle on XX, 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 Hˇ 4(X)\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 H¯ 2(X)\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 H¯ 2(X)\bar H^2(X) is an equivalence class in the category of smooth functors from P 1(X)P_1(X) to 1DVect1D\mathrm{Vect}.

(Here P 1(X)P_1(X) denotes some notion of paths in XX 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)H^3(X). Here P 1(X)P_1(X) is replaced by P 2(X)P_2(X) and 1-D vector spaces by something like 1-dimensional 2-vector spaces.

“And so on.”

In other words, H¯ n(X)\bar H^n(X) classifies abelian (n2)(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)U(1) in degree 3.

Let this 3-algebra be denoted by SPoin(11)\mathrm{SPoin}(11). Let the 3-algebra of an E 8E_8-Chern-Simons 2-gerbe be denoted by CSE 8\mathrm{CS}E_8.

Then we may try to study 2-gerbes with connection taking values in SPoin(11)CSE 8\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 8E_8-bundle. The 3-form part of the connection is automatically built from a Lie(U(1))\mathrm{Lie}(U(1))-valued part and the Chern-Simons 3-form of that E 8E_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 Inn(G 2)\mathrm{Inn}(G_2), for G 2G_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 Σ(Inn(G 2))\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 Σ(Inn(G 2))\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 ijd_{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
Read the post Differential n-Geometry
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
Read the post n-Curvature
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

Post a New Comment