What Is the Role of Twistor Space in Quantum Field Theory?

[atyy]

What is Nima talking about at 42 minutes?

http://www.cornell.edu/video/?videoID=913&startSecs=0

 

[atyy]

http://www.twistordiagrams.org.uk/
Twistor Theory and the Twistor Programme
Andrew Hodges
"Roger Penrose's books, The Emperor's New Mind (1989) and Shadows of the Mind (1994), refer in passing to twistor theory, but you would hardly guess from them that it has dominated his research work for nearly thirty years.
There is much more about twistor theory in his enormous new book The Road to Reality (2004). This book also contains some last-minute passages with the first discussion of Witten's new work and its implications for twistor theory."

http://arxiv.org/abs/0903.2110
The S-Matrix in Twistor Space
Nima Arkani-Hamed, Freddy Cachazo, Clifford Cheung, Jared Kaplan
"We recast the BCFW recursion relations in an on-shell form that begs to be transformed into twistor space. Our twistor transformation is inspired by Witten's, but differs in treating twistor and dual twistor variables more equally. In these variables the three and four-point amplitudes are amazingly simple; the BCFW relations are represented by diagrammatic rules that precisely define the "twistor diagrams" of Andrew Hodges. The "Hodges diagrams" for Yang-Mills theory are disks and not trees; they reveal striking connections between amplitudes and suggest a new form for them in momentum space. We also obtain a twistorial formulation of gravity."

 

[mitchell porter]

http://mitchell.physics.tamu.edu/Conference/string2010/
See videos, Thursday morning session.

 

[atyy]

http://mitchell.physics.tamu.edu/Conference/string2010/
See videos, Thursday morning session.

Thanks! Specifically session 1, I guess:
https://mediamatrix.tamu.edu/streams/327220/PHYS_Strings_2010_3-18-10A

 

[atyy]

http://pirsa.org/C10018
Space-time, Quantum Mechanics and Scattering Amplitudes
Mini course offered by Freddy Cachazo and Nima Arkani-Hamed

 

[atyy]

Not related, but I couldn't resist: http://arxiv.org/abs/hep-th/0311162!

I would really like to know how all this is related to AdS/CFT.

 

[negru]

Well this is about YM scattering amplitudes so of course it's related to ads/cft. One interesting find was the dual conformal invariance as coming from some fermionic T duality on the string side.
http://arxiv.org/abs/0807.3196

 

[mitchell porter]

AdS/CFT is a strong/weak duality, and these results are meaningful for weakly coupled SYM (since it's perturbation theory), so it ought to imply a connection between strongly coupled IIB string theory on AdS, and this Grassmannian object. Also, remember that this work descends from the twistor string, which is a topological string on a supertwistor space (https://www.physicsforums.com/showthread.php?t=415677"). So maybe the twistor string is the strongly coupled IIB string on AdS, but viewed under a duality transformation?

 

[negru]

I've also been wondering on how the twistor string fits into this. But so far the duality works only for tree level amplitudes...making it work for loop level would be pretty impressive.

But yeah I would find it hard to believe that the twistor strings and IIB strings are not related.

 

[atyy]

There's some loop level stuff in http://arxiv.org/abs/1008.2958.

 

[atyy]

There seems to be nothing relating twistorial S-matrix to AdS/CFT, except for the fact that the CFT is the same in both cases. I'm sure they're working on it.

http://arxiv.org/abs/1010.5009: "Recently the full N = 4 all loop integrand was proposed in a remarkable Yangian invariant form [12]. It would be interesting to see if the OPE limit can be applied directly at this level.[ 12] Also, it was recently understood how to generalize the bosonic null Wilson loops into super loops which are dual to amplitudes with arbitrary polatizations [13, 14]. It would be very interesting to apply the method we described to compute non MHV scattering amplitudes."

Till then, here's another entertaining paper: http://arxiv.org/abs/hep-th/0401094v2

And these transparencies are incredible! http://people.maths.ox.ac.uk/lmason/Tws/Penrose1.pdf

 

[chrispb]

There's something pretty deep going on here that I don't fully have my head around yet. I can tell you that there do exist uses for AdS/CFT in this line of research. Since what we're doing is computing scattering amplitudes in a CFT (N=4), we would ask what the corresponding objects are on the AdS side. The answer is stringy scattering amplitudes. Yes, weakly coupled amplitudes in the CFT become strongly coupled amplitudes in AdS. Recall that in N=4, the relationship is loops in AdS become the topological/nonplanar expansion in the CFT.

In addition, there is this dual (super)conformal symmetry floating around. On the CFT side of things, it can be interpreted as just some other generators that annihilate color-ordered partial amplitudes (but not individual Feynman diagrams). On the AdS side, though, Maldacena discovered a weak-weak duality that he called "fermionic T-duality" that interchanges particular amplitude computations in AdS with other computations. I don't know much about the AdS side of that, so you'd have to read that paper. I DO know that on the CFT side, if you consider the action of the fermionic T-duality, what it does is exchange amplitude calculations for Wilson loop calculations. Specifically, the dual group of the amplitude calculation gets mapped to the regular conformal group for the Wilson loop calculations and vice versa.

What was neat is the existence of both the regular and dual superconformal groups acting on amplitudes imply that there are an infinite number of generators that annihilate the N=4 partial amplitudes. Those generators span the "Yangian" people talk about.

At the loop level, some results are known. It is known that using dim reg to regulate the IR divergences breaks dual superconformal symmetry. However, there are other regulators that do not break the symmetry; the one I've heard the most about is the Higgs regulator. Seems obvious enough what it does. To the best of my knowledge, it is now known that N=4 amplitudes in the planar limit can be shown to all be invariant under the full Yangian, though I'm not 100% sure I believe that statement myself yet. My own work involves studying the nonplanar case, as well as searching for other theories that have these nice symmetries to facilitate writing down S-matrix elements.

Hope this helps!

 

[mitchell porter]

Hope this helps!

It certainly does! Summing up, it sounds as if this new symmetry, fermionic T-duality, is somehow the stringy counterpart of the Yangian symmetries in the CFT.

http://arxiv.org/abs/0912.3657" ) says a number of things which are worth highlighting here.

• Ordinary ("bosonic") T-duality: "From the point of view of the world sheet it is an abelian two-dimensional S-duality." (interesting as a conceptual connection between S and T duality)
• "Usual bosonic T-duality relies on using an isometry of the background to generate the T-duality transformation. Fermionic T-duality can be viewed as extending this idea to isometries of the fermionic directions in superspace. ... these are just the supersymmetry transformations, thus instead of using isometries to generate the T-duality transformations one uses the supersymmetries."
• Bosonic T-duality is a full symmetry of string theory, but fermionic T-duality is so far known to work only at tree level, and is broken at one-loop level - though this might just be an artefact of a particular algebraic approach.
• "Fermionic T-duality does not commute with bosonic T-duality" (this would be significant for the construction of an expanded U-duality group that includes fermionic T-dualities)

 

[mitchell porter]

http://motls.blogspot.com/2011/01/twistor-minirevolution-goes-on.html

 

[atyy]

Via Lubos's blog.

http://pirsa.org/11010111/

26:00 Is there a clock somewhere?

 

[atyy]

A review.

http://arxiv.org/abs/1104.2890
Scattering Amplitudes and Wilson Loops in Twistor Space
Tim Adamo, Mathew Bullimore, Lionel Mason, David Skinner
"...we instead seek to understand how to describe quantum field theory itself in twistor space. We shall see that this can indeed be done, and that twistor space scattering amplitudes and Wilson loops are beautiful objects in their own right, and are much more readily computable in twistor space than on space-time."