The String Coffee Table

Everybody is getting excited about Hitchin et al’s ‘generalized geometry’ (see also Luboš’ blog entry). I should hence hurry up with my PSM and Algebroids program, since that’s closely related.

Assume some closed string background preserving some susy. This comes with at least one Killing spinor whose square can be expanded into harmonic form fields ($J$, $\Omega$, $\phi$ being prominent examples in the Kähler, Calabi-Yau and G2 cases).

A brane in that background geometry preserves (part of) the super-symmetry if it is calibrated by those forms, that is if that form pulled back to the brane’s world volume is the volume form of that brane. This follows for example from the boundary conditions of the $S_\alpha$ fields on the string in the GS formulation.

However, on the brane there can be additional fields and I am specifically interested in the gauge field of D-branes. If the brane is a flat brane in a flat background, I know that the BPS condition is some sort of (generalized, non-linear) self-duality of the field-strength and this statement can be shown to be T-dual to the statement about calibrations.

My question now ist: What is known about the combined situation? What is the condition on a curved brane with gauge field in a curved background to preserve some susy?

Recently I mentioned here an idea by E. Akhmedov to construct a nonabelian surface holonomy using 2D TFTs. Today he has a new preprint on that issue.

On USENET David Roberts has mentioned some issues related with the desire to have weakened nonabelian bundle gerbes. It is not immediately clear what this would be, but the 2-bundle perspective might give a hint.

There seem to be interesting relations of the Poisson $\sigma$-model (PSM) and of Lie algebroids to all kinds of things that I am interested in. So I should begin to learn about them. Here’s a start.

‘What is known about the Segal-like formulation of open (topological) strings?’

I am aware of the following:

As pointed out on several other blogs, Joanna Karczmarek has been testing the waters with a downloadable version of the eprint arXiv. For the last few months, anyone with bittorrent installed on their machine has been able to download all of 2004’s papers in one go. But, as of yesterday, the whole shebang is now up for grabs.

You can grab the .torrent file either here or here. If you decide to download the arXiv, please let the bittorrent software continue to run, even after your download has finished. This makes the process faster and easier for everyone. In fact, it would be great if you just let the software run, in the background, for a few weeks. After all, if you’re on a University connection, you probably won’t even notice the 20 kB/s or so of bandwidth it uses.

By the way, the whole thing is only 7.4 Gb. That’s roughly a third of the smallest iPod on the market. So yes, you can carry the arXiv around in your pocket. The fun part, though, is how to index all of this data. It would be so boring if we just used the standard SLAC-type searches that we’re all used to. If you’re a windows user, you might want to check out Google Desktop. In a few weeks, Mac users who upgrade to Tiger can use Spotlight. Linux users have a few option as well, such as Beagle.

Update: As more people have started downloading the arXiv, the download speeds have really picked up. My download finished earlier today. One of the first things I noticed as I looked through the papers is that certain papers seem to be missing. My desktop environment generates small thumbnails of pdf files in place of icons, and I noticed that a few of the papers weren’t pdf files at all, but html with .pdf extensions.

Upon closer inspection, the html content of these files turns out to be the usual blurb that the arXiv offers up when it tries to convert a paper’s source to pdf. Clearly, these are the papers where the scripts failed to generate pdf. For instance, go check out hep-th/9108012 and try to grab a pdf version of that paper. After a few moments, the arXiv will return an error message, stating that it can’t generate a pdf file due to “incomplete or corrupted files”.

Not to worry, though. It seems as if the sources for some of these missing papers will produce valid pdf files with a minimum of fuss. For instance, if you download the source for hep-th/9108012 and pdflatex it, you’ll get a few errors. But you’ll also get a pdf version of the paper.

Lenny Susskind was recently here and discussed his recent paper on wormholes. As you probably know, he’s mentioned some of the counterarguments in his most recent paper. I think that there’s another possible perspective on the time difference. It’s sort of an intriguing combination of classical stuff and the uncertainty principle(s). I wrote it up formally, but it’s sketchy enough that I thought I’d just put it up here. Feel free to poke holes!

(Or I’ll just do it myself. I made a silly mistake in the last part. I’ve made some changes, and I’ll leave it up, but it’s late at night, and I can’t come up with a quick fix tonight.)

I am currently visting a CFT workshop at University of Bonn,

Lessons from low Dimensions - the many Aspects of Conformal Field Theory.

I am happy to be able to announce a new preprint:

J. Baez, A. Crans, U. Schreiber & D. Stevenson

From Loop Groups to 2-Groups

math.QA/0504123.

Abstract:

We describe an interesting relation between Lie 2-algebras, the Kac–Moody central extensions of loop groups, and the group $\mathrm{String}(n)$. A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the ‘Jacobiator’. Similarly, a Lie 2-group is a categorified version of a Lie group. If $G$ is a simply-connected compact simple Lie group, there is a 1-parameter family of Lie 2-algebras $\mathfr{g}_k$ each having $\mathfr{g}$ as its Lie algebra of objects, but with a Jacobiator built from the canonical 3-form on $G$. There appears to be no Lie 2-group having $\mathfr{g}_k$ as its Lie 2-algebra, except when $k = 0$. Here, however, we construct for integral $k$ an infinite-dimensional Lie 2-group $\mathcal{P}_k G$ whose Lie 2-algebra is equivalent to $\mathfr{g}_k$. The objects of $\mathcal{P}_k G$ are based paths in $G$, while the automorphisms of any object form the level-$k$ Kac–Moody central extension of the loop group $\Omega G$. This 2-group is closely related to the $k$th power of the canonical gerbe over $G$. Its nerve gives a topological group $|\mathcal{P}_k G|$ that is an extension of $G$ by $K(\mathbb{Z},2)$. When $k = \pm 1$, $|\mathcal{P}_k G|$ can also be obtained by killing the third homotopy group of $G$. Thus, when $G = \mathrm{Spin}(n)$, $|\mathcal{P}_k G|$ is none other than $\mathrm{String}(n)$.

[Update: I am aware of that problem with the incorrectly-displayed TeX code above. I am hoping to find the solution to that problem soon.]

A while ago I had started thinking about 2-NCG. Watching me trying to define vector 2-bundles in that context people (Aaron, that is) made me look into the derived category description of D-brane configurations.

Like Zaphod B. looking into the total perspective vortex I found myself nothing but confirmed in my basic idea (of course possibly due to a similar loss of sense of reality…).

The result is a little sketch: 2-NCG – A Suggestion.

Have a look and decide for yourself if this is an April Fool’s joke or not.