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March 22, 2007

Split Real Forms

The science blogosphere has been all atwitter, this week, about E 8E_8, and a purported breakthrough in the representation theory thereof. Most of the posts were not particularly informative. The best of the lot was here, on our sister blog, the n-Category Café.

Not having much intelligent to say, I thought I would take a pass on adding to the frenzy. But, on reconsideration, I thought I might, at least, add ϵ\epsilon about the connection with physics.

First of all, you have to realize what is being talked about is not our friend, the compact Lie group, E 8E_8, but a distant cousin, the split real form, which I will, henceforth, denote by E˜ 8\tilde{E}_8. A complex, simple Lie algebra, 𝔤 \mathfr{g}_{\mathbb{C}}, can have several real forms, only one of which is the Lie algebra of a compact Lie group, GG. At the opposite extreme is the split real form, whose corresponding Lie group, G˜\tilde{G} is “as noncompact as possible.” For example, so(2n,)so(2n,\mathbb{C}) has a compact real form SO(2n)SO(2n), and a split real form, SO(n,n)SO(n,n) (and intermediate real forms, SO(2nk,k)SO(2n-k,k)).

Anyone familiar with the heterotic string will recognize the “E” series of compact Lie groups: E 8,E 7,E 6,E 5=Spin(10), E 4=SU(5),E 3=SU(3)×SU(2) \begin{gathered} E_8,\; E_7,\; E_6,\; E_5=Spin(10),\\ E_4=SU(5),\; E_3=SU(3)\times SU(2) \end{gathered}

The corresponding split real forms E˜ 8,E˜ 7,E˜ 6,E˜ 5=Spin(5,5), E˜ 4=SL(5,),E˜ 3=SL(3,)×SL(2,) \begin{gathered} \tilde{E}_8,\;\tilde{E}_7,\; \tilde{E}_6,\; \tilde{E}_5=Spin(5,5),\\ \tilde{E}_4=SL(5,\mathbb{R}),\; \tilde{E}_3=SL(3,\mathbb{R})\times SL(2,\mathbb{R}) \end{gathered} appear in the maximal supergravity theories (the dimensional reductions of 11 dimensional supergravity down to d=11nd=11-n dimensions). Specifically, the scalars in the supergravity multiplet take values on the homogeneous space E˜ n/K n\tilde{E}_n/K_n, where K nK_n is the maximal compact subgroup1 of E˜ n\tilde{E}_n. K 8=Spin(16),K 7=SU(8),K 6=Sp(4), K 5=Spin(5)×Spin(5),K 4=Spin(5),K 3=SU(2)×SO(2) \begin{gathered} K_8 = Spin(16),\; K_7 = SU(8),\; K_6 = Sp(4),\\ K_5= Spin(5)\times Spin(5),\; K_4=Spin(5),\; K_3 = SU(2)\times SO(2) \end{gathered}

By construction, E˜ n\tilde{E}_n acts a a global symmetry group of the supergravity theory.

Alas, with the exception of the d=3d=3 (and possibly d=4d=4) cases, the supergravity theory is nonrenormalizable, and must be UV-completed. The completion is Type-II string theory (or M-theory) compactified on a torus. The higher dimension operators in the dd-dimensional effective Lagrangian are not invariant under the continuous E˜ n\tilde{E}_n, but only under a discrete E˜ n()\tilde{E}_n(\mathbb{Z}) subgroup, called the U-duality group.

There are massive BPS states in the theory, and these can be organized into representations of E˜ n()\tilde{E}_n(\mathbb{Z}). If you are interested in studying the spectrum of such BPS states (say, to write down a U-duality-invariant formula for the entropy of blackholes in this theory), then you are interested in the representation theory of E˜ n\tilde{E}_n.

For d=3d=3, that’s E˜ 8\tilde{E}_8, and that’s presumably where these latest results might hold some interest for physicists.

1 Looking at this table, I suspect I am not being sufficiently careful about the centers of the respective groups.

Posted by distler at 12:25 PM | Permalink | Followups (5)

March 16, 2007

Validator Face-Off

Can someone please point me to an online Validation Service that deals correctly with the following following three documents? They are not, by any stretch of the imagination, torture tests. They’re about as simple a trio of examples as I could cook up.

But I’ve yet to find an online Validator that handles all three correctly (so far, I’ve only found one that handles two of the three correctly).

Posted by distler at 10:52 AM | Permalink | Followups (9)

March 12, 2007

The Standard Model Landscape

One of the most consistently misunderstood features to emerge from the study of string compactifications is the existence of a “landscape” of metastable vacua. Surely, it is a grave defect of the theory to possess so many solutions! Obviously, the string theorist must be on the wrong track.

But, if you think about the matter, you quickly realize that the essential ingredients: coupling to gravity, and a source of violation of the null-energy condition, are rather commonplace. Arkani-Hamed, Dubovsky, Nicolis and Villadoro have a beautiful paper, in which they point out that the requisite conditions are present in the rather minimalist context of the Standard Model, coupled to gravity, with massive neutrinos. As long as there are no other light fields1, their analysis holds for any theory containing the Standard Model, including — one hopes — string theory.

In Type II flux vacua, it is negative tension of the orientifold planes that supplies the violation of the null energy condition. In Arkani-Hamed et al’s story, it is the Casimir energy due to light fermions with periodic boundary conditions on a circle. In the simplest case, they looks at a compactification on AdS3×S 1\times S^1 with positive 4D cosmological constant. At the classical level, the potential for the “radion”, the 3D scalar governing the radius of the S 1S^1, is of a runaway form, and the solution, just described, is a funny way of writing dS4.

But the Casimir effect, for fermions with periodic boundary conditions on the S 1S^1, induce a 1-loop correction to the potential for the radion, which tends to make the circle shrink, rather than expand.

By happy coincidence, the neutrino masses in the real world are comparable in scale to the 4D cosmological constant, and can stabilize the radion at a finite radius of the S 1S^1 (of order a few microns). The 3D effective cosmological constant is negative, with the AdS3 radius somewhere on the order of the 4D Hubble length (10 25\sim 10^{25}m).

The existence of these vacua is entirely an infrared effect. The existence of more massive degrees of freedom like, say, the electron, induce only tiny corrections of order e m e/m μe^{-m_e/m_\mu}. Indeed, the smallness of these corrections means that the scalar dual to the photon in 3D has a nearly flat potential.

For all intents and purposes, particle physics in any of these vacua is the same as in our 4D world.

There is quite a fun story to do with other “compactification” geometries.

But, I think the bottom line is that we should not be particularly alarmed at the presence of a large number of vacua. Any theory worthy of our consideration will, likely, possess a similarly large number of vacua. If we should be disturbed by anything, it is that, out of the plethora of string vacua found to date, none of them looks sufficiently like our world, rather than that there are too many that do.

1 The presence of other light fields like, say, the axion, certainly affects the analysis and could upset the vacua that they found, or could create even more vacua, depending on the details.

Posted by distler at 1:45 AM | Permalink | Followups (6)

March 11, 2007

BZR Feeds

Since my last post, I’ve gone on to make a number of bugfixes and minor feature-additions to my branch of Instiki. The main changes are listed below the fold.

But, thanks to Morten Frederiksen, there is now an easier way to keep track of develoments in itex2MML and Instiki. The corresponding BZR repositories now have atom feeds (itexToMML, Instiki).

Posted by distler at 11:41 PM | Permalink | Post a Comment

March 1, 2007


If you’re gonna spend a week or two, immersed in writing software, you ought to have something to show for it. My previous post didn’t quite qualify. It was all about not having something (malicious javascript) to show.

Regular readers probably know about my dissatisfaction with Keynote. It’s a crappy vehicle for writing a talk containing equations. There’s no way that I should have to switch applications just to type an equation. And it’s completely absurd that I should have to fiddle around placing a little PDF image of an inline equation, to get the baselines to line up with the surrounding text. Yes, there are some utilities that make the situation slightly less intolerable. And, yes, it’s better than PowerPoint (which really is setting the bar awfully low).

But, really, one ought to be able to do better. S5 is a very nice presentation software package based on HTML and Javascript. Andrea even incorporated a nice S5 input-syntax in Maruku. Unfortunately, the Javascript that drives S5 isn’t compatible with real XHTML. So, at first blush, that seemed like a non-starter.

On the other hand, I’ve quite some experience fixing web software that’s broken when served as real XHTML

Turns out that it wasn’t that hard.

So, I hereby announce the advent of S5 support in Instiki. An S5 slide show is just another Wiki page, with slightly-modified input semantics. Just set the category of your page

:category: S5-slideshow

and it doubles as an S5 slideshow. An “S5” View button appears at the bottom of the page. Click on it to start your slide show.

Of course, all the standard goodies are still there: itex equations, rendered to MathML, inline SVG, …

No spinning pie-charts. But, I suppose, if you really want them, they, too, are possible (or will be).

There’s a sample slide show (click on the “S5” link to start the show).

Now it’s time to get back to posting about Physics…

Posted by distler at 4:38 PM | Permalink | Followups (15)