## Thursday, February 17, 2011

### String theory in exotic R^4

Today in arxiv there is a very curious article:Quantum D-branes and exotic smooth R^4 written by Torsten Asselmeyer-Maluga, Jerzy Krol.

The article is actually the second part of a previous one: Exotic smooth R^4 and certain configurations of NS and D branes in string theory.

The abstract of the first (in date order) article reads:

In this paper we show that in some important cases 4-dimensional data can be extracted from superstring theory such that a) the data are 4 Euclidean geometries embedded in standard $\mathbb{R}^{4}$, b) these data depend on NS and D brane charges of some string backgrounds, c) it is of potential relevance to 4-dimensional physics, d) the compactification and stabilization techniques are not in use, but rather are replaced. We analyze certain configurations of NS and D-branes in the context of $SU(2)$ WZW model and find the correlations with different exotic smoothings of $\mathbb{R}^{4}$. First, the dynamics of D-branes in $SU(2)$ WZW model at finite $k$, i.e. the charges of the branes, refers to the exoticness of ambient $\mathbb{R}^{4}$. Next, the correspondence between exotic smoothness on 4-space, transversal to the world volume of NS5 branes in IIA type, and the number of these NS5 branes follows. Finally, the translation of 10 dimensional string backgrounds to 4 Euclidean spaces embedded as open subsets in the standard $\mathbb{R}^{4}$ is achieved.

I still haven't had time to read the full article, but it looks quite interesting, and beautifully, specially from the perspective of someone whose favourite area of maths is topology. The idea is to see if someone can make string theory in a background of R^4 with a different differential structure than the usual one. One of the most amazing discoveries of differential topology was that there were different differential structures for R^4 than the usual one. That means that although like a topological manifold R^4 is unique there are different differentiable manifolds that are compatible with it's topological structure. An explicit characterization of that exotic structures in terms of coordinate is difficult and their existence is proved by means of topological techniques. In the paper it is made use of h-cobordism.

Later he begins the string theoretical construction, using SU(2) WZW (wess-zumino-witten= CFT's, NS 5 branes, D branes, etc. AS I still haven't read the article carefully, nor the continuation of it (the today's arxiv article ) wouldn't give more details. Only to say that it looks like a very intriguing area of research.

Also today in arxiv there is a very interesting, but much more conventional article of Miche Dine: Supersymmetry from the Top Down whose abstract is:

If supersymmetry turns out to be a symmetry of nature at low energies, the first order of business to measure the soft breaking parameters. But one will also want to understand the symmetry, and its breaking, more microscopically. Two aspects of this problem constitute the focus of these lectures. First, what sorts of dynamics might account for supersymmetry breaking, and its manifestation at low energies. Second, how might these features fit into string theory (or whatever might be the underlying theory in the ultraviolet). The last few years have seen a much improved understanding of the first set of questions, and at least a possible pathway to address the second.">If supersymmetry turns out to be a symmetry of nature at low energies, the first order of business to measure the soft breaking parameters. But one will also want to understand the symmetry, and its breaking, more microscopically. Two aspects of this problem constitute the focus of these lectures. First, what sorts of dynamics might account for supersymmetry breaking, and its manifestation at low energies. Second, how might these features fit into string theory (or whatever might be the underlying theory in the ultraviolet). The last few years have seen a much improved understanding of the first set of questions, and at least a possible pathway to address the second.

The article is quite pedagogical, and even begins with an ultra fast introduction to supersymmetry. Certainly recommendable.

On the subject of supersymmetry in the LHC era I thing that everybody would must read the last entry of Jester's blog: What LHC tells about SUSY that discussed the paper of the ATLAS collaboration: Search for supersymmetry using final states with one lepton, jets, and missing transverse momentum with the ATLAS detector in sqrt{s} = 7 TeV pp.

Well, certainly the expectations of Lubos of an early discovery of SUSY in the LHC are gone, but still there are good reasons to be patient, as explained by Lubos himself or by the Dine's paper.

By the way, while writing this entry I have seen that Lubos himself has written an entry about the Dines paper, you can read it here. At the moment of writing my entry he hasn't given many details about the article, but possibly he will edit his post and discuss the paper in more detail.

Update: Lubos has read this entry and has written a very intersting essay about the general relevance (or irrelevance) of the pathological mathematical structures in physics .

About the actual series of papers in exotic R^4 he doesn't say too much because he claims that he doesn't understand the paper. I have been studying the subject, including some references, and I am still going on. Much of the mathematics (differential topology: h-cobordism, topological surgery, tubular neighbourhoods) are familiar to me, but some more recent concepts are new to me. Still I think that I can follow the general argumentative line of the mat part. I get somewhat more loose in other points of the WZW model in an SU(2) background, but still I think that I get the general argumentation. As soon as I end reading a few more references I'll try to expose the key ideas.

Anyway, there is a difference here with the case commented but Lubos. R^4 is the only R^n that admits different smooth structures for the same topological structure. In that aspect it is not a case of searching for a pathology but a case where the pathology appears by itself. In fact most people who have heard about that particularity of R^4 have always though that maybe that could be the ultimate reason that we live in a four dimensional manifold. Of course what they lacked is a way to relate that peculiarity of R^4 to any actual physic. This people seem to have advanced somewhat in that direction, but as far as I understood they are far of the objective (if that is their objective, that probably it is not the case).

Mitchell said...

Asselmeyer-Maluga has written a whole series of highly repetitive papers trying to connect her favorite exotic R^4 construction to physics. But it's done on an extremely superficial basis - as I recall, in the case of the alleged connection to D-branes, the "connection" is little more than the same group showing up in both constructions. My recommendation, if you want to understand the paper, is to remember that it is actually about two completely different things. One is this construction from exotic differential geometry, the other is something from string theory, and almost certainly there simply is no relationship between the two of them, despite what the paper says.

Javier said...

Hi Mitchell, thanks for your comment.

I read the two articles I linked and as you say the connection is weak

Maybe the best point is that I watched the references they gave and I arrived to the book "exotic smoothness and physics" that looks quite interesting, at least in it´s mathematical side.

Anonymous said...

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