Cabinet de Curiosites dot com - STRINGS 04

June 30, 2004

Cabinet de Curiosites dot com - STRINGS 04

Posted by Urs Schreiber

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The annual string theory conference is taking place at College de France in Paris this year. Fortunately this is a little closer to where I live than Tokyo, so that I don’t have to miss it. Refreshed by a marvelous weekend with my girlfriend at the banks of Seine I am now switching from right to left brain hemisphere and listen to up to eleven plenary talks per day about that theory which once fell from here into the 20th century and which is now being pulled back to the ortochronous frame in order to catch up with the accelerating cosmic expansion.

While this is still work in progress my humble task shall be to accelerate the expansion of my personal horizon.

In the age of internet communication one interesting aspect of conferences is always the identification and meeting of e-pen pals. I was glad to meet Jacques Distler and Robert Helling in person for the first time, after quite a while of virtual acquaintance.

In case you haven’t seen it, first check Jacques’ musings (I, II) on $Strings_{04}$ which were produced close to real-time and where of course much more erudite comments on the plenary talks can be found than I am able to produce.

In fact, I’ll only mention the first of today’s talks, which was by Ashoke Sen on 2 D-string theories. As opposed to many other talks wich were concerned with model building, this one stood out as one that nicely addressed the ‘big picture’ of string theory, albeit just in a toy model. The main point was to show how the continuum worldsheet description maps in detail to the Matrix Model point of view. One crucial technique used by Sen was the correspondence between rigid target space gauge symmetries to conserved D-brane charges. This works as follows (assuming that I recall the details correctly):

A rigid gauge transformation in closed string field theory is generated by a ghost number 1 gauge parameter field $\Lambda$ , which, since it corresponds to a rigid transformation, has vanishing momentum. Assume more generally that such a field has some fixed momentum $p_0$ . Construct a 1-parameter family of fields $\Lambda(p)$ such that $\Lambda(p_0) = \Lambda$ . BRST invariance of $\Lambda$ can then be expressed in terms of some string field $\phi(p)$ as

(1)

(Q_B + \bar Q_B)|\Lambda(p)\rangle = (p- p_0)|\Phi(p)\rangle \,.

Next consider some boundary state $\langle \mathca{B}|$ describing some brane. Using the BRST invariance of $\langle \mathca{B}|$ one sees that

(2)

\langle \mathcal{B}| [(c_0 - \bar c_0), Q_B + \bar Q_B] |\Lambda\rangle = (p-p_0)\langle \mathcal{B}| (c_0 - \bar c_0) |\Lambda\rangle \,.

But this expression vanishes identically, because the ghost 0-modes are not saturated:

(3)

(p-p_0)\langle \mathcal{B}| (c_0 - \bar c_0) |\Lambda\rangle = 0 \,.

Fourier transforming this expression by introducing the object

(4)

F(x) = \int dp\, e^{-ipx} \langle \mathcal{B}| (c_0 - \bar c_0) |\Phi(p)\rangle

it is equivalent to

(5)

\nabla \cdot \left(e^{i p_0 x } F(x) \right) = 0 \,.

But this tells us that there is a conserved quantity $e^{i p_0 x } F(x)$ for every gauge parameter field $\Lambda$ at fixed momentum.

Sen compares these conserved quantities with those arising in the Matrix Model of 2D string theory and finds lots of interesting equivalences. But the details are beyond the scope of my notes and my remaining time this evening.

Posted at June 30, 2004 8:29 AM UTC

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2 Comments & 0 Trackbacks

Re: Cabinet de Curiosites dot com - STRINGS 04

Hello Urs, it is Friday 9:47 now, Paris time. So I think that in a few minutes you will be on position to confirm or deny my pessimistic view about the new life of twistor theory.
Is Witten just reissuing his 1986 paper, or it is going to be the new string trend for a couple years?

My worry is if this “take on” could become accidentally connected to the other objects from Penrose, spin-networks, then subsuming the LQG theory into string theory.

Posted by: alejandro rivero on July 2, 2004 8:48 AM | Permalink | Reply to this

Re: Cabinet de Curiosites dot com - STRINGS 04

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Hi Alejandro -

it’s too bad - but I can’t listen to Witten’s talk. Due to teaching duties I had to return home Thursday night. This is a real pity, since there are lots of interesting talks today, like that by Berkovits on pure spinor formalism. Let’s ask Jacques Distler about today’s talks instead.

I don’t know much about the details of twistor strings, but it should be emphasized that this is part of a general trend of better understanding of field theory in general and $N=4$ SYM in particular, by relating it to string theory.

In fact, a good fraction of talks at ${Strings}_{04}$ was devoted to rather pure gauge theory topics. For instance there was the talk by Gopakumar on how to identify the string dual of the planar limit of any gauge theory by examination of the path integral, a talk by Dixon on recent progress in the computation of QCD amplitudes by means of using similarities with $N=4$ SYM (for instance tree level amplitudes in both theories are identical), and a series of talks on recent progress in computing anomalous operator dimensions in SYM and the comparison with AdS/CFT, by Tseytlin, Zarembo and Beisert. Then there was the impressive talk by Aharony on deconfining transition in gauge theories in finite volume.

So I don’t see why one should be pessimistic when it is demonstrated that twistor strings provide yet another stringy description of gauge theory. Of course the way twistors appear here may be morally different than what Penrose originally intended them for.

Posted by: Urs Schreiber on July 2, 2004 9:09 AM | Permalink | PGP Sig | Reply to this

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June 30, 2004