Saturday, January 29, 2011 ... //

PI: Nima Arkani-Hamed on twistor uprising

Mikael has pointed out a video of a fresh 100-minute talk (100 minutes is equal to one decimal Nima hour) that Nima Arkani-Hamed gave at the Perimeter Institute three days ago. Before I began to watch, I wasn't sure whether I would finish.

Well, I became much more certain about the answer during the talk. ;-)

The title uses David Gross's favorite term "uprising" instead of my "minirevolution" and if you watch the talk, you may figure out whether the twistor minimirevolution has been downgraded or upgraded. ;-)

Nima explains that the goal is to think different - to eliminate the word "Feynman" from the QFT calculations as completely as possible. ;-) In particular, the new description and calculational methods proudly make locality in the ordinary spacetime obscure while they succeed in making many other, more exotic properties of the N=4 gauge theory manifest.

As has been known for some years, the twistor variables simplify the maximally-helicity-violating and other scattering amplitudes. In recent years, it became clear that they also make the dual superconformal symmetry manifest - and the dual superconformal symmetry, together with the ordinary superconformal symmetry, generate the infinite-dimensional Yangian symmetry.

The dictionary that translates some basic spacetime and momentum space concepts to the twistor space is sketched, together with some geometric interpretation in terms of polygons. The momentum conservation becomes non-manifest as well. The quantity that is conserved (the momentum) is not linear but bilinear in the twistor fields - so it may be understood as an orthogonality. Using the space of k-dimensional planes in an n-dimensional space, a Grassmannian, one may parameterize all the possible orientations of the twistors etc.

Nima conjectures that all the amazing simplifications of the integrand, integral, and results that emerge from this formalism indicate that there is a new description of the whole AdS/CFT system - something that he called T (for "twistor") on his diagram but I will call it T-theory.

The rest of this article was posted on January 4th, 2011 at 10:14 am Prague Winter Time

Twistor minirevolution goes on

At the end of 2010, a Princeton-Perimeter-Oxford group has released two new preprints about the miraculous simplification that Roger Penrose's twistors bring to the calculation of scattering amplitudes of the maximally supersymmetric gauge theory in 3+1 dimensions:
A Note on Polytopes for Scattering Amplitudes (24 pages)

Local Integrals for Planar Scattering Amplitudes (84 pages)
The lists of authors of both papers include Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo (now working for peRIMeter), and Jaroslav Trnka (yes, a Czech name!). The first paper was also co-written by Andrew Hodges, a pioneer of the gay liberation movement from the 1970s who also happens to be a top Oxford mathematician. ;-)

The short paper plays with some geometry and shows that various key mathematical objects that appear in both papers (and many previous papers) have a nice geometric interpretation in terms of areas or volumes of various polygons and polytopes that you may construct in the twistor space - a complex projective space - and its Cartesian powers.

It could have been written because of some previous insights by Andrew Hodges and his extensive knowledge of the geometry relevant for the twistors. Those mathematical suggestions looked confusing and overly abstract to physicists such as Arkani-Hamed but as you can see, the situation has changed.

The long paper is a serious paper about physics and scattering amplitudes which is why no pure mathematician is among the co-authors. It sheds a completely new light on the sequence of developments that we have seen during the last decade.

History: going back to the 1960s

In 1967, Roger Penrose proposed twistors as a fundamental tool for the physics of spacetime. He argued that they should be relevant for quantum gravity. For decades, this claim remained nothing else than a wishful thinking but in the 2000s, the statement turned out to be likely to be true even though as of January 2011, the most well-established applications of twistors remain unrelated to gravity as a force.

Twistors are closely related to spinors, objects that may be understood as "square roots of vectors". I like to say that twistors may similarly be interpreted as "square roots of spacetime points".

In 2+2-dimensional spacetimes (ours is 3+1-dimensional), twistors may also be identified with purely light-like 2-dimensional planes in spacetime. Two such planes generically intersect in a light-like line - and the light-like line may actually be identified with the twistor itself. In this dictionary, spacetime points become lines in the twistor space.

If you switch from the twistor space to the spacetime, objects of higher dimensions become objects of lower dimensions, and vice versa. The envelopes - or lines/planes connecting several points/lines - get mapped to the intersection of the higher-dimensional objects, and vice versa. It's a lot of fun.

To calculate with the twistors, one has to realize their close relationship with the spinors. In the modern treatment, a light-like momentum "p" in 3+1 dimensions is written as "lambda^c.lambda*^d", a tensor product of two 2-component spinors. One does additional procedures we may very superficially sketch later. But because I probably won't, let me say that the Fourier transform over these "lambda" variables (square roots of the momenta) is important, too. And it is very useful to transform over the right-handed "lambda*" variables only - in a left-right-asymmetric way; only in this way, the true power of twistors emerges.

But between 1967 and 2003, for more than 35 years, twistors were only used to study abstract mathematics and in the context of physics, they were only good for an unusual description of free (non-interacting) massless fields - aside from a few exceptions with solutions to non-linear equations related to instantons etc. No generic interactions were allowed, however; no genuine dynamics which is the "bulk of physics" could have been studied by twistors.

Witten enters the scene

In 2003, Edward Witten published his papers on the twistor treatment of the maximally supersymmetric Yang-Mills theory in four dimensions. For the first time, geometry in the twistor space was used to calculate scattering amplitudes - quantities knowing about some real dynamics and interactions in physics.

The scattering of N gluons (or their superpartners) in the gauge theory only occurs if the points describing the gluons in the twistor space - which replaces spacetime or momentum space and should be viewed as "something in between them" - belong to the same (complex) line. Well, that's the case if the polarizations of the gluons are "maximally helicity violating" (MHV).

What is it? Well, relabel all gluons in a scattering process so that all of them are incoming. The right-handed outgoing ones become left-handed incoming ones and the left-handed outgoing ones become right-handed incoming ones. What is the MHV amplitude?

If all the incoming gluons are left-handed (i.e. if all left-handed gluons changed to fully right-handed ones) then... well, then the scattering amplitude vanishes - a fact that is pretty nontrivial by itself. This amplitude is "more than more than allowed" helicity-violating one so it has to vanish. You won't be able to prove the vanishing "immediately" using the Feynman rules; it's not just about the angular momentum because the angular momentum is not quite the same thing as helicity (notice the variable axes of the helicity!). Witten's twistor prescription for the scattering amplitude makes the vanishing manifest.

You must add some right-handed particles for the amplitude to be nonzero. If all the gluons are left-handed and one of them is right-handed, then the amplitude... vanishes again. ;-) It's the "more than allowed" helicity-violating amplitude.

Fine, I don't want to try your patience. If all of them are left-handed but two are right-handed, the amplitude is nonzero. It's the first nonzero amplitude in this sequence so we call it the MHV, the maximally helicity-violating amplitude. (It only makes sense to study it for 4+ gluons.) Similarly, with three right-handed gluons, you would get the NMHV (here, "N" stands for "next-to-"), and so on.

In late 2003, Witten proposed a description for the MHVs in terms of lines on the twistor space and the NMHVs and NNMHVs etc. in terms of higher-degree curves on the twistor space. He has also framed all his prescriptions in terms of a topological string theory defined on a twistor space that kind of worked but so far, it hasn't been shown too useful and - apparently - hasn't quite known about the newest developments. But maybe it's just because no one has been able to deal with Witten's topological twistor string theory properly.

2004: Disconnected rules

Meanwhile, in 2004, Cachazo, Svrček, and Witten replaced the curved submanifolds of the twistor space - to calculate the NMHVs, NNMHVs, and friends - by the sum over several diagrams, each of which uses straight (complex) lines in the twistor space only.

This new picture made it much easier to derive and check various recursion relations for the amplitudes - what happens if you add a new particle to the process, and so on. Also, particle physicists are arguably much more trained in summing many diagrams than in working with a single curved manifold in algebraic geometry so the "CSW" transition has become very popular while the higher-degree curves in the twistor space slipped into silence.

(We have argued - with Andy Neitzke and Sergei Gukov - that the disconnected CSW recipe is equivalent to the connected Witten's curved recipe because integrals on both sides get localized on intersecting lines e.g. degenerate versions of the curved submanifolds in both cases.)

Imagine lots of work related to the CSW rules and various extensions during a few years.

New structures and symmetries in N=4 SYM

Meanwhile, another minirevolution, the BMN minirevolution, has begun to overlap with the twistor revolution sketched above. What is (or was) the BMN minirevolution? In 2002, Berenstein, Maldacena, and Nastase gave new life to another old invention by Penrose, namely his Penrose limit of geometries (also known as the pp-waves which is almost the same thing).

The Penrose limit of the AdS5 x S5 space was translated to a specific new refinement of the 't Hooft limit on the gauge theory side (where the R-charge of the "long" operators also scales properly with the square root of the number of colors) and it allowed the people to check that the gravitational, AdS side of the AdS/CFT correspondence contains not only gravity but also all excited string modes (and other objects) and their interactions as predicted by string theory.

So the AdS/CFT is not just a vague correspondence between a gravitational theory and a field theory: it has been known for a long time that the gravitational side (and therefore also the field-theoretical sides) contains all the special new stringy objects, exactly as they were predicted by string theory. Obviously, the gauge theory is consistent so the equivalent quantum gravitational theory has to be consistent as well. String theory is the only consistent theory of quantum gravity which implies that it had to be the full string theory on the AdS side if the correspondence works at all. (Of course, this fact was pretty much known as soon as Maldacena wrote his correspondence down: he realized that the conjecture was right by analyzing some technical details of black hole entropy calculations in string theory which worked better than expected.)

The BMN pp-wave minirevolution was interesting because it led the people to calculate complicated amplitudes in the N=4 gauge theory and see some patterns and hidden symmetries that are not obvious from the Lagrangian - and that don't exist in less supersymmetric gauge theories.

A big portion of these new symmetries may be summarized as the "Yangian symmetry". This symmetry appears in the planar amplitudes - according to 't Hooft's classification of the gauge-theoretical diagrams. It's some kind of a U(1)^{infinity} symmetry - an infinite number of new conserved charges you didn't expect. In the AdS/CFT dictionary, the planar diagrams get translated to tree-level diagrams in string theory for the AdS space (Riemann surfaces without any "handles") even though they typically contain many ordinary loops of the gauge-theoretical Feynman diagrams.

Now, the planar limit is pretty much equivalent to the tree-level string theory in the AdS space which contains the same information as the classical limit - classical string field theory in the AdS5 x S5 space, if you wish. The latter is classically integrable: you should be able to write all relevant correlators etc. for infinitely many fields of the string field theory (it's just mostly free theory with many fields but even the interacting terms are classically "manageable") in terms of rather elementary analytic functions. So the corresponding description of the same physics on the gauge-theoretical side should also be integrable and the Yangian symmetry produces the infinitely many new charges that make the theory integrable in the field-theoretical variables and that allow you to calculate the results without much work.

Experts in this subdiscipline began to get familiar with many unusual patterns and coincidences that hold for all planar N=4 gauge-theoretical scattering amplitudes and the explanation of these patterns and coincidences. Meanwhile, that's what the twistor people were doing, too. The two minirevolting cultures began to merge. The spin chain (the integrability business that evolved from the BMN breakthrough) folks should better learn some twistors and vice versa.

2008: dual superconformal symmetry, fermionic T-duality, etc.

Things didn't stop in recent years. On the contrary. As TRF readers learned from my reports from Strings 2008, the year brought us a rather extensive new stringy knowledge that has helped to explain the special form of many amplitudes.

Take the non-linear sigma-model - world sheet theory describing strings propagating on a curved background - for the AdS5 x S5 space. Read the paper by Alday and Maldacena, perform some T-duality on the four coordinates that belong to the boundary - where the CFT lives (and forgive me that they're not compact as T-duality usually expects) - and combine it with a totally new, 2008-fresh "fermionic T-duality" by Berkovits and Maldacena. Remarkably, you get the same model back while you may learn totally new things.

So the AdS string theory is kind of self-dual under an unexpected generalization of T-duality. The resulting "dual spacetime" is very cool because the theory is superconformally invariant in this "dual spacetime" as well. More precisely, the planar limit of the ordinary AdS5 x S5 string theory is "dual-superconformally invariant" - as originally pointed out by Drummond, Henn, Koršemský, and Sokačev - while the stringy loop corrections add some rather controllable "violations" of this symmetry.

All this stuff is very funny and powerful because the planar amplitudes themselves may be expressed as the expectation values of purely light-like Wilson loops living in the dual spacetime. Moreover, the dual spacetime on the CFT side has a very simple description, too: it is almost literally the Fourier transform of the original spacetime.

Use symbols "x_1, x_2, ... x_k" for the positions of "k" gluons living in the ordinary four-dimensional spacetime. The theory is translationally invariant so the correlator only depends on the coordinate differences "x_i-x_{i+1}" - including "x_k - x_1". A funny trivial fact about these "k" coordinate differences is that their sum identically cancels and vanishes.

The coordinate differences "x_i-x_{i+1}" play a very simple role in the dual spacetime: they're interpreted simply as the momenta "p_i". Note that two sentences ago, I just explained that the momenta add up to zero.

The twistor-space-based expressions for the scattering amplitudes have made the new symmetries - dual superconformal symmetry and/or the Yangian symmetry - more transparent. However, the twistor space is not diffeomorphic to the spacetime in any sense so generically, the twistor space descriptions are non-local (the association of the interactions with particular points in the spacetime is not self-evident from the form of the amplitudes). Because they're also based on chiral spinors, they like to totally obscure the left-right parity symmetry of the N=4 gauge theory!

2010: isolating the integrand with the beautiful properties

In the newest papers, Arkani-Hamed et al. do a great new step because they write the known twistor-based formulae for the scattering amplitudes as integrals over a universal domain - a Grassmannian (the space of M-dimensional subplanes of an N-dimensional space). By finding the residues, you may decompose the integral into several "seemingly topologically different" Feynman diagrams.

It's incredible that all the diagrams for a scattering process may be written as a single integral - a single diagram. Ordinary quantum field theory usually forces you to sum an exponentially (or factorially) large number of Feynman diagrams to obtain the amplitude for a complicated process. Of course, we know another framework in which many Feynman diagrams of field theory arise from a single diagram: perturbative string theory. In this sense, the twistor expression has similar "unifying properties" as perturbative string theory.

(The connected recipe for twistors, in terms of higher-degree curves in the twistor space, was pretty much giving a single integral as well. However, its correct generalizations to many loops isn't quite known these days - which may only be because the key researchers in the contemporary twistor business don't like curved algebraic geometry too much.)

The rules to produce the amplitudes as the unified twistor integrals could even be a "version of string theory" although the formalism that unifies these two formalisms is not known at this moment.

The rewriting of the amplitudes as a universal single integral has many advantages. The integrand actually makes both the old-fashioned locality as well as the new unusual Yangian symmetry manifest! Parity remains obscure - as in all twistor approaches.

It also removes one of the big obstacles that have always discouraged me from studying loop amplitudes in the twistor language at all: in gauge theory, the amplitudes are infrared-divergent so they're not really well-defined and it's not clear what you're calculating. So why should you do it too accurately?

However, many people were saying it didn't matter - as I also emphasize (e.g. in futile debates with Vladimir), the infared divergences were a "real insight of physics" resulting from your having asked a wrong question, not a sickness of a theory. And there was a set of well-defined finite observables behind these amplitudes. These people turned out to be right in this twistor case, too. Because Arkani-Hamed et al. can write the amplitudes as integrals, it actually turns out that the integrand itself is finite and all the infrared divergences arise from including extreme corners of the integration domain - much like the "tau = i.infinity" region of one-loop stringy diagrams (thin tori).

While the physical interpretation of the integrand is not quite clear at this point, it makes sense to argue that it is a physically meaningful object satisfying many physical criteria and constraints. And it is finite.

2011+: understanding the "twisting beast" in between AdS and CFT?

If the progress continues, people could eventually find a complete definition of this twistor-based description for any amplitudes and prove its equivalence with the perturbative Yang-Mills theory on one side, and perhaps the AdS string theory on the other side, as well as all of its desired symmetries (superconformal, dual superconformal, parity, Yangian). What are its degrees of freedom and their interactions? Are there some new types of string theory or spin chains or something else?

If that's true, the "twisting beast" could turn out to be an extremely useful intermediate diplomat who helps us to prove the AdS/CFT symmetry in an explicit way.

And it could tell us much more than that.

I have more conceptual things to say about these relationships (many of which could be wrong, vacuous, or well-known) and analogies but because I am afraid that they wouldn't be appreciated, let me stop at this point. This twistor business, while highly technical, is hiding some nontrivial wisdom that knows about hidden symmetries of supersymmetric theories. And it could perhaps shed some new light of the "full string theory" including AdS5 x S5 gravity (e.g. off-shell gauge theory - note that only on-shell gauge theory is studied in this whole business so far) and perhaps even more general mysteries.

And that's the memo.