Gauge theory, string theory, and twistor theory converge

[mitchell porter]

http://online.kitp.ucsb.edu/online/qcdscat11/

You can see it happening in these talks. For now it's just d=4 N=4 super-Yang-Mills and d=4 N=8 supergravity, but there is every reason to think that the relationships being discovered there will be extended (in more complex forms) to other gauge theories such as the standard model.

AdS/CFT revealed that a gauge theory could be a string theory. The twistor string revealed that a gauge theory could be simpler as a twistor theory. Arkani-Hamed et al rewrote planar N=4 theory (only planar Feynman diagrams) in terms of a new object, a Grassmannian in twistor space, with an extremely simple definition. Zvi Bern in his talk says you should be able to get non-planar N=4 (the rest of N=4 SYM) by transforming planar N=4, and then from there it's just a trivial step to N=8 supergravity. And behind it all is the theory of motives, and other aspects of the mathematical universe created by Grothendieck and others. In his talk, Marcus Spradlin exhibits how a 17-page formula for a "remainder function" (part of a scattering amplitude) was reduced to a single line using the motivic theory of polylogarithms, and says that probably the whole theory (N=4 SYM again, in this case) is a Hopf algebra. The latter perspective was already developed by Connes and Kreimer when they studied renormalization theory, so I take all of this to mean that there is an algebra of diagrams (Feynman diagrams and other diagrams) in which amplitudes are derived from a combinatorial motivic cohomology. Yet somehow these quantities are also equivalent to volumes of polytopes in AdS space - this talk by Maldacena might be a useful complement to the KITP talks:

http://pirsa.org/10040040/

So, in short, quantum field theory, the workhorse of particle physics for at least sixty years, turns out to have a hidden side. There was already a big conceptual revolution in QFT in the 1970s, one that still isn't in the textbooks - the revolution associated with Wilson's renormalization group, which made conformal field theories secretly important to the structure of QFT. This revolution looks to be at least as big. Of course, it has been brewing for years, at least since AdS/CFT (which took off in 1998), and the revival of twistor theory dates from the end of 2003. But now it looks to be reaching a new culmination, perhaps the complete solution of d=4 N=4 super-Yang-Mills, in terms of a motivic twistor theory. It's amazing to watch it happening.

 

[atyy]

Hodges: Harmony is projective momentum space! (Well, the other part about the lines being representations of logs, we do know about.)

"That brings out the usefulness of notation and representation. ... We are looking for something that will represent the music of gauge field scattering amplitudes in an equally marvellous way."

 

[atyy]

http://online.kitp.ucsb.edu/online/qcdscat11/mason/

At 3:30 , someone in the audience brings up another disadvantage of twistor space - something about 4, I think - is he referring to the number of spacetime dimensions?

 

[mitchell porter]

I think the objection is not just that twistor space is only useful for four-dimensional theories, but that it's only useful for scale-invariant theories. I'm led to wonder what relationship there is between the way you introduce mass in twistor space, and the way you introduce mass in AdS/CFT (both of which provide a description of N=4 super-Yang-Mills).

Lionel Mason's sample calculation looks both simple and deep. I wonder if we should try to reproduce it here? The action he uses can be found in equations 3.2 and 3.4 of http://arxiv.org/abs/hep-th/0604040" .

 

[atyy]

I think the objection is not just that twistor space is only useful for four-dimensional theories, but that it's only useful for scale-invariant theories. I'm led to wonder what relationship there is between the way you introduce mass in twistor space, and the way you introduce mass in AdS/CFT (both of which provide a description of N=4 super-Yang-Mills).

Lionel Mason's sample calculation looks both simple and deep. I wonder if we should try to reproduce it here? The action he uses can be found in equations 3.2 and 3.4 of http://arxiv.org/abs/hep-th/0604040" .

That's probably be useful for some.

But I'm just starting at a much more elementary level - have just checked out from the library Huggett and Tod's Introduction to Twistor Theory, which is recommended in Adamo et al's review http://arxiv.org/abs/1104.2890 .

A couple of other references, not so directly related, but which seem easy to read are Dunajski's http://arxiv.org/abs/0902.0274 , and Krasnov's http://arxiv.org/abs/hep-th/0311162v1 , which comments "It is not too often emphasized, however, that an analog of twistors exists in any spacetime dimensions. Moreover, there is an analogous concept even for the spaces Rn with the usual flat positive definite metric. These analogs are well known to mathematicians."

 

[negru]

You also have twistors in AdS

 

[qsa]

This was posted few months back(#96)

https://www.physicsforums.com/showthread.php?p=2978247&highlight=kerr#post2978247

here is part of it

Not to keep you all in suspense before I continue, ENERGY is nothing but the length of this line (actually 1/L), which is nothing but your usual momentum K, although here it appears geometrically. All interactions (forces) arise naturally from simple logical relationships of these lines belonging to different particles. My website has not included many new findings including the famous 1/r law, but can be seen from fig.2. which mimics Hydrogen 1s energy level. In some respect, no energy(momentum) means no space defined.

This theory goes very well with Smolin’s comment that particles as end of lines should be studied and Joakim’s(google) linking entropy (verlinde’s) to the wavefunction and twister theory with Kerr which considers particles as end of lines.


this theory is based on Buffon's needle which is linked to Radon transform which is the non-complex brother of Penrose transform(twister). no need for complex because time des not enter directly but can be calculated.

 

[mitchell porter]

The place where I'm now looking for immediate progress in understanding (my understanding, that is) is the relationship between the calculation of the "BDS remainder function" at weak coupling and at strong coupling. An attempt to set out the issues:

We are in d=4 N=4 super-Yang-Mills. There is a duality between http://arxiv.org/abs/1010.1167" . Page 2 of the linked paper makes it look rather simple: the 1-loop correction to the tree-level scattering amplitude has a "dual conformal symmetry" in its momenta. At the level of string theory - that is, in the dual AdS description of N=4 SYM - this can be explained, or at least exhibited, by "fermionic T-duality", which is a self-duality of the AdS5 space and which maps (planar) N=4 SYM onto itself. Momentum space maps to a new position space, and the Wilson loop dual to a scattering amplitude is a null polygon with lightlike edges which correspond to the momentum 4-vectors of the external particles in the scattering amplitude.

So this is the first basic fact to cling to: A scattering process maps to a Wilson loop in a dual space-time. Geometrically it's very simple: the Wilson loop is drawn by appending the 4-vectors to each other, just like in ordinary vector addition, and conservation of 4-momentum means that the loop will close on itself. But the point is, not just that we can construct a geometric figure in the dual space, but that the QFT calculations in the two pictures (scattering, loop) coincide.

Next step: N=4 SYM has a coupling constant. At strong coupling, we can compute the expectation value of the Wilson loop in AdS space - it turns out to be equal to the area of the minimal surface in AdS space which ends on the loop (which exists on the boundary of AdS space). This is the subject of Maldacena's talk, linked in my comment #1. He says it is exactly the same as the "soap bubble problem", of determining the area of a soap bubble, but the surface is warped towards the center of AdS because of the negative curvature.

At weak coupling, you use perturbation theory, and that means all these twistorial techniques, as well as BCFW recursion relations (which I think derive from the twistor string representation of gauge theory, anyway).

The actual expectation value is divergent. More precisely, it is the sum of a divergent part; the BDS (Bern, Dixon, Smirnov) ansatz; and the remainder function, which is a conformal invariant. The remainder function is zero for anything less than a six-point function because there are no conformal invariants for five points or less (here, I'm just reciting what I've read, I haven't looked into these invariants at all, though I think they were also showing up in Mason's twistor talk).

I gather that when the calculation (six-point two-loop amplitude, or the corresponding Wilson loop) is performed, these three parts show up, both at weak and at strong coupling.

In Maldacena's talk, as well as going over the steps needed to compute the amplitude (which here, remember, equals the area of a surface in AdS bounded by the polygonal Wilson loop), which involves integrability and a formal application of the "thermodynamic Bethe ansatz" which is mysterious because there's no temperature in sight. The final formula he obtains - for the remainder function, I presume - is a function of "cross ratios", simple combinations of the 4-vectors (external momenta of the scattering amplitude / edges of the polygonal Wilson loop).

At weak coupling, the piece de resistance of Spradlin's KITP talk is a simplification of the remainder function to a formula one line long, again a function of the cross ratios, in which special functions called polylogarithms show up. The one-line formula, as I previously mentioned, was found by taking a 17-page formula computed using known methods (including twistor techniques), finding something like a character of the formula using Goncharov's motivic technique, the "symbol map", and then determining a polylogarithmic expression with the same character (the same "symbol"). (And then a few undetermined constants were determined by much simpler considerations.)

So the state of the art, if one is looking for a unified perspective on the computation of this amplitude - unified from weak to strong coupling - is that on both sides you have that same tripartite decomposition, and the interesting part is the remainder function, which is always a function of the cross-ratios, but at strong coupling you get it by applying integrability theory to the geometric AdS picture, while at weak coupling there is a magical and dramatic simplification of the brute-force computation which can be exhibited by these motivic methods, but which can't be directly motivated by any known property of the theory. And the challenge is to interpolate between these two perspectives and these two calculations.

 

[Physics Monkey]

You can see it happening in these talks. For now it's just d=4 N=4 super-Yang-Mills and d=4 N=8 supergravity, but there is every reason to think that the relationships being discovered there will be extended (in more complex forms) to other gauge theories such as the standard model.

I'm genuinely curious, do you really believe this is true? I'm not trying to be overly pessimistic, but I don't see a lot of evidence so far that these techniques will have wide application. On the other hand, I see a great deal more evidence for wide application of ads/cft (although still flimsy evidence by some very reasonable standards). Perhaps I'm simply poorly informed, so I wonder what you had in mind when saying this?

Of course, I do appreciate the vast simplification that appears for the very special theories considered.

 

[negru]

Just one reason why all of this should work for non susy theories is that it's connected to Zvi Bern's work on the color-kinematic duality (BCJ conjecture)
http://arxiv.org/abs/1004.0693

which they claim works for non-susy and for all loops.And anyway even if it doesn't work beyond n=4 ym it's still pretty good, because anything you find in n=4 ym must mean something for string theory

 

[atyy]

In Arkani-Hamed's talk he does say he really doesn't know whether strings, AdS/CFT and twistors are 3 different special aspects of a more comprehensive theory T, or whether twistors are in fact T. He does say it's wonderful high energy theorists finally have something that doesn't happen in crappy piece of metal (http://online.kitp.ucsb.edu/online/qcdscat11/arkanihamed/ @26 min)

Also, isn't N=8 supergravity supposed to be in the swampland (which I think means it isn't a consistent theory by itself, and has no consistent completion in strings - could it have a consistent completion in twistors)?

 

[suprised]

Also, isn't N=8 supergravity supposed to be in the swampland (which I think means it isn't a consistent theory by itself, and has no consistent completion in strings - could it have a consistent completion in twistors)?

N=8 supergravity does have a completion in strings, which is the low energy limit of type II strings on the six-torus.

The issue is whether one can decouple all the extra stringy stuff (massive modes eg), ie remove the "completion", and still obtain a consistent "pure" supergravity theory. There are indications that one cannot do this (http://arXiv.org/abs/0704.0777). On the other hand there are indications that pure N=8 supergravity may be perturbatively finite. However there could be inconsistencies beyond perturbation theory, so some kind of non-perturbative completion may be necessary, nevertheless.

I guess it's fair to say that right now it is unclear whether pure N=8 supergravity is a consistent theory by itself, but at any rate there exists a consistent completion of it in string theory (which per def amounts to adding extra stuff).

 

[atyy]

We are in d=4 N=4 super-Yang-Mills. There is a duality between http://arxiv.org/abs/1010.1167" .

Maybe not? http://arxiv.org/abs/1103.3008

 

[mitchell porter]

Maybe not? http://arxiv.org/abs/1103.3008

Some sort of duality exists, but it might not be true to all orders. Or maybe no-one found the exact right form of it yet. Meanwhile, in http://arxiv.org/abs/1103.3714" , those same authors propose to extend the duality to a "triality" (see page 3) involving correlators as well as amplitudes and Wilson loops.

While I'm listing still more connections that I don't know what to do with, I have to mention http://arxiv.org/abs/1004.4735" . An elementary extrapolation says one will also exist for (0,2) theory, but what I want to know is, will there be a twistorial equation for M-theory as a whole?

Furthermore, I was just claiming https://www.physicsforums.com/showthread.php?t=485800&page=6#94"... So it looks like some enormous synthesis of twistors, M-theory, and Wilsonian QFT is conceivable; but the details...? We just have to wait for the "reconceptualization flow" to arrive at a fixed point.

 

[atyy]

Shouldn't the attractor of that final flow be more interesting? - like a limit cycle or strange attractor;)

 

[mitchell porter]

I think there's a further stage of this convergence happening now, summarized in the introduction of this paper.

AdS/CFT was introduced with the observation that there ought to be a duality between string theory in AdS4, AdS5, and AdS7, and various boundary CFTs. AdS holographic dualities for other bulk dimensions have been discovered, but those three are the prototypes. The prototypical dual theory for AdS5 is N=4 super-Yang-Mills, a theory which has been known for 30 years. The prototypical dual theory for AdS4 is ABJM theory, a theory of "super-Chern-Simons plus matter" that was discovered in 2008, but whose general nature was anticipated by John Schwarz a few years before. Finally, we don't know what the boundary theory for AdS7 is, but it should have some relationship to the six-dimensional strongly coupled theory that has been discussed a lot in recent years.

What's on display in today's paper, are unexpected connections between physical calculations in ABJM theory and N=4 Yang-Mills; and also connections with N=8 supergravity, which is certainly part of the twistor revival, even if no-one knows how or whether it fits into string theory. I see this as progress towards discovery of the true nature of M-theory (understood broadly, as the theory underlying string theory in all its forms). M-theory is the bulk theory in all the AdS examples, so this isn't quite the same as the mid-1990s unification of string theories, but it's showing that M-theory calculations in different spaces are producing the same result, for unknown reasons. This implies that a further leap in understanding lies ahead.