## July 21, 2008

### T-Duality and Dual Superconformal Symmetry

One of the most interesting talks at the Simons Workshop in Stony Brook was by Nathan Berkovits, about his joint work with Maldacena. Roughly a year ago, Alday and Maldacena made a striking observation about the strong coupling behaviour of scattering amplitudes in $\mathcal{N}=4$ Super Yang-Mills at large-$N$. Namely, that they possess a conformal symmetry in momentum space, and can be computed using a sort of “T-dual” version of AdS/CFT. One studies Wilson loops in momentum space, which are polygons consisting of $n$ light-like edges (momentum conservation is the statement that the polygon closes). These can be computed, as the areas of a minimal surface (string worldsheet, with boundary on the polygon) in the bulk (“T-dual”) $\text{AdS}_5\times S^5$. I blogged about their work before, so I’ll refer you to my post, and the paper, for more details.

What Nathan and Juan do is provide an explanation of what T-duality is supposed to be taking place here.

The first oddity is that we are T-dualizing in noncompact directions (the four coordinates of $\mathbb{R}^{3,1}$). This is a very funky thing to do, and surely would not be sensible at higher genus. But the planar limit should be OK.

On the sphere,we need to require that none of the vertex operators carries momentum. On the disk, we can abide open string vertex operators which carry momentum. In the T-dual theory, these are open strings “stretched” between different D-branes. $\mathbb{R}^{3,1}$ was the world volume of the D3-branes. T-duality turns Neumann boundary conditions in these directions into Dirichlet. So the computation “really” involves $n$ open strings connecting D($-1$)-branes. This is not quite a Wilson loop, but close enough for Government work.

Most importantly, since the bulk theory is AdS, T-dualizing in the $\mathbb{R}^{3,1}$ directions (alone) would produce a non-constant dilaton, which varied logarthmically with the AdS radial direction. To “fix” this, Berkovits and Maldacena T-dualize as well, in 8 fermionic coordinates (in the pure spinor formulation of the Green-Schwarz string). That is, they follow the Buscher procedure, not just for bosonic translational symmetries, $x^i\to x^i+ a^i$, but also for fermionic translations, $\theta^a \to \theta^a + c^a$. This produces another, cancelling, contribution to the dilaton gradient. The net result is a “T-dual” $\text{AdS}_5\times S^5$ background, with constant dilaton.

What’s more remarkable is that, since we had to choose an (anti-commuting) set of 8 supercharges, out of 32, on which to T-dualize, we seem to have manifestly broken the $SO(6)$ R-symmetry. The end result, however, must be $SO(6)$ invariant.

To complete the story, they need to conjecture a precise relationship between the “T-dual” open string scattering amplitude computation and the Wilson line computation, valid in the $\alpha'\to 0$ limit. in fact, they do better than that and conjecture, in an appendix, a general formula for the open string disk amplitudes in this formalism.

Very interesting stuff, and — I hope — an important step in understanding many of the mysteries of $\mathcal{N}=4$ SYM scattering amplitudes from AdS/CFT.

#### Update:

As always, I’m slightly behind the curve. There’s also a very nice paper, today, by Beisert et al, covering much of the same material. It concentrates a bit more on the concrete realization of the dual superconformal symmetry.
Posted by distler at July 21, 2008 11:43 PM

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### Re: T-Duality and Dual Superconformal Symmetry

Hi - You may not be aware of this but in fact we had a paper out two weeks ago precisely on the subject of dual superconformal symmetry of scattering amplitudes in N=4 SYM: arXiv:0807.1095. In it we found dual superconformal symmetry as a natural generalisation of dual conformal symmetry.

Just to recall the context, dual conformal symmetry was found first as a property of the loop integrals entering the perturbative expansion of gluon scattering amplitudes. It also arises at strong coupling for gluon scattering amplitudes after the bosonic T-duality as discussed by Alday and Maldacena in their paper from May last year. Their work shows that this symmetry is connected with the ordinary conformal symmetry of light-like Wilson loops in N=4 SYM and that, at strong coupling, there is an intimate connection between scattering amplitudes and such Wilson loops. Surprisingly, this connection was found to hold also at weak coupling (see arXiv:0707.0243 and arXiv:0707.1153 for the start of the story).
In a couple of papers from last year, based on this newly found relation to Wilson loops, we derived the consequences of dual conformal symmetry for scattering amplitudes - it is implemented by an anomalous Ward identity (anomalous due to the infra-red divergences present in scattering amplitudes).

Up to this point, dual conformal symmetry was only a symmetry of the loop corrections to MHV amplitudes - the new development from our paper of two weeks ago was that it has a natural extension to act on all amplitudes (not just MHV) including the helicity structures (which for MHV is just the tree-level amplitude). This extension necessarily extends dual conformal symmetry to dual superconformal symmetry and we give a general conjecture as to how it is realised on all scattering amplitudes (to all loops). We were able to explicitly verify it for the first case of six-point NMHV amplitudes at tree-level and one loop.

Today’s papers by Berkovits and Maldacena and Beisert, Ricci, Tseytlin and Wolf give some very nice insight into how dual superconformal symmetry arises from the string sigma model. In particular the doubts about whether dual conformal symmetry was confined to the strong coupling regime now seem to have gone (which is a good thing in view of the accumulated evidence from weak coupling that Wilson loops and scattering amplitudes are indeed related). As you pointed out in your post this is related to the cancelling contributions to the dilaton after bosonic and fermionic T-duality. As far as I am aware (although I am happy to be corrected) they stop short of explaining how the dual superconformal symmetry is actually realised on scattering amplitudes from the sigma model perspective. We gave a realisation of the dual superconformal algebra in our paper and conjectured how it is realised on the amplitudes, but to see this from the string theory I guess that one would have to understand better how the vertex operators behave.

Also, as you rightly indicated, it would be very interesting to understand the relation between Wilson loops and amplitudes in this wider context - this might give a strong clue as to how to generalise the Wilson loop/amplitude duality beyond MHV amplitudes to include the whole lot.

Sorry for the long comment, but I thought it would be a shame if these developments were missed in the discussion on your blog.

Regards,

James

Posted by: James on July 22, 2008 11:20 AM | Permalink | Reply to this

### Re: T-Duality and Dual Superconformal Symmetry

Thanks for the reference. As always, I am way behind the curve.

In particular the doubts about whether dual conformal symmetry was confined to the strong coupling regime now seem to have gone …

It’s definitely still restricted to the planar limit. But, I agree that the evidence is that it does extend to finite ‘t Hooft coupling.

As far as I am aware (although I am happy to be corrected) they stop short of explaining how the dual superconformal symmetry is actually realised on scattering amplitudes from the sigma model perspective.

In an appendix to their paper, Berkovits and Maldacena propose a formula for the $n$-point scattering amplitude, computed from this T-dual theory. In some sense, that formula is the answer to your question.

Sorry for the long comment, but I thought it would be a shame if these developments were missed in the discussion on your blog.

Seems to me like a good way to start off the discussion. Thanks!

Posted by: Jacques Distler on July 22, 2008 11:47 AM | Permalink | PGP Sig | Reply to this

### Re: T-Duality and Dual Superconformal Symmetry

If I read the paper correctly the formula in the appendix only refers to the flat target space case and is in this version of their paper not derived from anything (although it does seem to give the right results for the flat space case). Really deriving scattering amplitudes in the pure spinor formulation of AdS5S5 seems to be hard.

A related note (which I gleaned/misappropriated from James’ work) is that the full amplitude is actually *not* dual conformal invariant: the helicity structures prevent this. In the MHV case this is simply the tree level amplitude not being invariant. So somewhere in the Berkovits and Maldacena story this ‘anomaly’ must arise, but I don’t see where and if I remember Berkovits’s talk in Paris correctly, neither do they.

Posted by: Rutger on July 23, 2008 2:38 AM | Permalink | Reply to this

### Flat

If I read the paper correctly the formula in the appendix only refers to the flat target space case and is in this version of their paper not derived from anything (although it does seem to give the right results for the flat space case).

I’m not sure what you mean. This is an open-string scattering amplitude, and the boundary, $\mathbb{R}^{3,1}$, is flat.

Really deriving scattering amplitudes in the pure spinor formulation of AdS5S5 seems to be hard.

Agreed. The formula in question is just conjectured, not derived. Still, it passes many consistency checks…

Posted by: Jacques Distler on July 23, 2008 8:22 AM | Permalink | PGP Sig | Reply to this

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