## June 3, 2007

### Connections on String-2-Bundles

#### Posted by Urs Schreiber

Aspects of this were discussed here:

$\;\;$From loop groups to 2-groups

$\;\;$On the String 2-Group

$\;\;$ Higher Gauge Theory and Elliptic Cohomology

$\;\;$ Chern Lie $(2n+1)$-algebras

$\;\;$ The second edge of the cube (higher Cartan connections)

Posted at June 3, 2007 3:46 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1300

### Re: Connections on String-2-Bundles

I see you’ve adopted a super-terse style now Urs. “How to get a spinning string from here to there in under sixty seconds” :-)

Posted by: Bruce Bartlett on June 3, 2007 4:46 PM | Permalink | Reply to this

### Re: Connections on String-2-Bundles

I see you’ve adopted a super-terse style now Urs. “How to get a spinning string from here to there in under sixty seconds” :-)

The idea was:

First give a rough idea about why a connection on a String-bundle should be a 2-functor that locally takes values in the String-2-group.

Then pass to the differential picture of that and discuss morphisms from the tangent algebroid to the String and/or the Chern-Simons Lie $n$-algebra.

The diagrams towards the end will require quite some time to explain, I think.

But maybe I should add more paragraphs as in section 2 “to be skipped on first reading” which give more details.

Posted by: urs on June 3, 2007 5:36 PM | Permalink | Reply to this

### Re: Connections on String-2-Bundles

Nice stuff!

But what is this document supposed to be, Urs? Notes for a talk? A sketch of a paper you’re writing? A treat for café customers only?

I’m interested in part because my talks at the Abel Symposium and the Erwin Schrödinger Institute this summer will cover rather similar material (though not as much of it).

By the way, you refer to a paper I’m writing with Danny. If you want a title for that paper, you can use “A Classifying Space for 2-Bundles”.

Posted by: John Baez on June 4, 2007 6:54 PM | Permalink | Reply to this

### Re: Connections on String-2-Bundles

Nice stuff!

Thanks!

But what is this document supposed to be, Urs?

It’s been written up for the talk I give on Wednesday in our elliptic workshop.

It’s main purpose is to organize my thoughts on what I will say. I intend to roughly follow the narrative of these notes.

Another intended purpose is as a reference for others. I have made what I thought was a good experience with passing around notes of this kind and then going through these in the talk. That’s the way I did it with my previous $\mathrm{String}(n)$-talk last year. I had the impression that it was a good strategy with topics like this, which have a heavy accent both on a nice bird’s eye story but also on lots of technical constructions.

(In this respect I am not satisfied with the present notes, in that they omit too many details, especially towards the end. I ran out of time a little when writing them.)

I’m interested in part because my talks at the Abel Symposium and the Erwin Schrödinger Institute this summer will cover rather similar material

Feel free to use/cite whatever you like. I’d be honored.

By the way, you refer to a paper I’m writing with Danny. If you want a title for that paper, you can use “A Classifying Space for 2-Bundles”.

Good, thanks! I’ll include that. Can’t do so right now, though, because the dork that I am left the LaTeX source on my home PC, to which I have no access over the week.

Will also have to fix some other references (like that to Anders Kock’s work) which currently just appear as placeholders.

And I also wanted to include a reference to our paper-in-preparation on the “canonical 2-rep”. This, too, will now have to wait until the weekend.

(By the way, Alissa said something about possibly visiting me in Hamburg in July for finishing that paper. But I am not sure when exactly. I’ll be on vacation for two weeks in the middle of July.)

(though not as much of it).

Even though there is already a lot of stuff in these notes, a glaring omission is that I don’t discuss how to impose the Cartan conditions on these String and Chern-Simons connections that appear towards the end. This will involve finding total spaces of string bundles characterized by “principal” actions of the Lie algebra of derivation of the Baez-Crans Lie algebra.

I was looking for literature on this for the ordinary case, but didn’t find any. Maybe you can help me:

given a space $P$ and a Lie algebra $g$ and an “action” of $g$ on $P$ in the form of a Lie algebra morphism $g \to \Gamma(T P) \,,$ I want to say what it means for this action to be “principal”.

I think I know how to do this, but this must be in the literature somewhere.

Posted by: urs on June 4, 2007 7:29 PM | Permalink | Reply to this

### Re: Connections on String-2-Bundles

Not quite sure what you are asking:
principal is usually in the context of
a bundle
so for your $g\to \Gamma(TP)$
you would want to have some notion of
vertical’ tangent vectors and the action
of g on each fibre’ should be the standard

or is the problem precisely removing the
` ’
above?

Posted by: jim stasheff on June 12, 2007 1:33 AM | Permalink | Reply to this

### Re: Connections on String-2-Bundles

Not quite sure what you are asking:

We need to reformulate the fact that the action $R : P \times G \to P$ on a bundle $p : P \to X$ is principal equivalently in terms of properties of its differential $R_* : \mathrm{Lie}(G) \to \Gamma(T P) \,.$

What I thought is this:

that the action is free should translate into $R_*(\xi) \in \Gamma(T P)$ being a nowhere vanishing vector field, for all $\xi \in \mathrm{Lie}(G)$.

That the quotient $P/G$ is base space $X$ should translate into the fact that $\mathrm{Im}(R_*)$ spans the kernenl of $p_* : T P \to T X$ pointwise.

So, my idea was that principality of the action should be “pointwise” exactness of this sequence $0 \to \mathrm{Lie}(G) \stackrel{R_*}{\to} \Gamma (T P ) \stackrel{p_*}{\to} \Gamma (T X) \to 0 \,.$

Maybe what I really mean by that is exactness of $0 \to P \times \mathrm{Lie}(G) \stackrel{R_*}{\to} T P \stackrel{p_*}{\to} T X \to 0$ for each fixed $p \in P$.

Or, what I actually mean is maybe ordinary exactness of $0 \to \mathrm{Lie}(G) \stackrel{R_*}{\to} \Gamma (T P ) \stackrel{p_*}{\to} \Gamma (T X) \to 0$ but regarded as a sequence of $C^\infty(X)$-modules.

That’s why I was asking for literature. I had the feeling that this is the way to go, but felt unsure about how to formulate it best.

Posted by: urs on June 12, 2007 11:11 AM | Permalink | Reply to this
Read the post Extended Quantum Field Theory and Cohomology, I
Weblog: The n-Category Café
Excerpt: On understanding extended quantum field theory and generalized cohomology.
Tracked: June 8, 2007 2:25 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 5, 2007 11:01 AM
Read the post Higher Gauge Theory and Elliptic Cohomology
Weblog: The n-Category Café
Excerpt: A talk on higher gauge theory and its relation to the work of Stolz and Teichner.
Tracked: August 1, 2007 9:30 AM
Read the post Obstructions for n-Bundle Lifts
Weblog: The n-Category Café
Excerpt: On obstructions to lifting the structure n-group of n-bundles.
Tracked: September 12, 2007 11:32 PM

Post a New Comment