Sunday, July 29, 2012

Permutations join twistor minirevolution

I have finally found the time to watch Nima Arkani-Hamed's excellent Strings 2012 lecture.
Scattering Amplitudes and the Positive Grassmannian (four formats, 40 minutes)

(Flash video, PDF only)
I recommend you the Flash video; the PDF file without Nima's words and handwaving is vastly less comprehensible, it seems to me.

Let's hope that this toasted twister is an OK symbol of Nima's talk. It's similar to snack wrap chicken in McDonald's

You may be pretty sure that the talk gives us the hottest information about the state of the twistor minirevolution.

The readers may want to watch it as many aspects of the BCFW and similar formulae that looked very complicated get clarified. Suddenly they have a reason. Lots of arbitrariness disappears. Different, equivalent diagrams boil down to the same invariant structure that the physicists are starting to see in front of their eyes and touch by their hands.

So what are the points you shouldn't miss while watching the talk?

First of all, the papers in 2010 or so would include lots of diagrams with black and white vertices. Those diagrams may have had many loops – but they will be interpreted in a new, very simple way: one diagram is a permutation.

Let us begin with a question we should have understood for some years: Why do the diagrams have black and white vertices? What does this racial segregation mean? Well, note that all these vertices are cubic. The black vertices correspond to \((++-)\) while the white vertices correspond to \((--+)\) where the signs denote the left-handed or right-handed helicity of the external gluons that we scatter. You may say that the white vertices and the black vertices are left-right mirrors to each other. One of them is the maximally helicity-violating process (MHV) and the other one is "one step even more than maximally" helicity-violating process.

Another important point is that all the "propagators" in these twistor diagrams are on-shell. Nevertheless, they still contain everything you need to emulate the usual off-shell Feynman diagrams. An added "bridge" and therefore a new loop shifts the spinors in the right way.

But this new development is a story about permutations that play a key role.

We should first understand: What are we permuting? Well, if we're scattering \(n\) gluons, we are just permuting them. So there are \(n!\) different permutations. A funny thing is that the gluons transform in the adjoint of \(SU(N)\) which means that they have another predetermined permutation – encoded in the trace of the color matrices. So we're adding another permutation \(p\) into the game.

Now, let's return to the old physics in 1+1 dimensions. If one lives in the Lineland, he may talk about the "ordering of the people" along the Lineland. If you permute two adjacent points on a line, and every permutation may be decomposed into a product of such transpositions, it's an actual objective operation; the two points must penetrate through one another. In a process where \(n\) particles scatter in a 1+1-dimensional world, you find the Yang-Baxter equations that must hold for the "matrices" remembering the internal operation on the exchanged particles. Those equations have important applications in braid groups, knot theory, and other things. They effectively tell you that the ordering of the transpositions shouldn't matter as long as the overall permutation is the same.

However, these things look like special features of 1+1-dimensional physics. In 3+1 dimensions, for example, it seems impossible to "order points". However, in a specific sense, this research by Nima Arkani-Hamed and various collaborators – mathematicians and physicists – is an extrapolation of the "laws of the Lineland" to a 3+1-dimensional world.

How do we find out the permutation from the black-and-white Feynman-like diagram?

This is a somewhat key point of the talk so you shouldn't miss it. You have those \(n\) external gluons distributed along the circle and they're connected to something like a Feynman diagram with black and white cubic vertices inside. What is \(p(j)\), i.e. the gluon assigned to the \(j\)-th gluon by the permutation associated with the diagram?

The rule to find out \(p(j)\) is straightforward. Place Mickey Mouse on the \(j\)-th gluon and send him inside i.e. through the Feynman diagram with simple instructions. Whenever the Mickey Mouse hits a black vertex, he turns left; when he hits a white vertex, he turns right. (I hope that it doesn't matter too much if I reverted the rules upside down but you surely need to use their conventions if you want to check the detailed formulae.) Finally, Mickey Mouse gets out of the maze: he reaches another external gluon.

It's not hard to see that the resulting rules \(p(j)\) for each \(j\) define a permutation. Each black-or-white cubic vertex becomes a clockwise or counter-clockwise roundabout. Mickey Mouse always gets out on the next exit so whenever he arrives from different roads to the roundabout, he leaves via different roads, too. That probably implies that \(p(j)\) is simple.

It must also be an "onto" map as long as we can prove that Mickey Mouse always gets out of the maze "somewhere", to another external gluon. But that's necessary the case, too. He can't get caught into loops. If he could, there would be the first edge \(E\) (piece of road) that he visits for the second time. But the roundabout before this edge is a one-to-one map so it's not possible for two different "previous edges" to be sent to \(E\).

It makes sense. One may associate a permutation with those Feynman-like black-and-white cubic graphs. Now, the next big statement is that the amplitude only depends on the permutation. There are actually many equivalent ways to write down an amplitude but as long as they are associated with the same permutation, they are evaluated to the same result.

Well, there seem to be some subtleties that Nima described and I am not sure whether I fully understood them. They prefer to write the permutation of \(n\) external gluons in such a way that \(p(j)\) is always greater than \(j\). So if you have a definition where \(p(j)\) is smaller than \(j\), you redefine \[

p(j)\to p(j)+n.

\] Fine. However, they also want to generalize the permutations into decorated ones (technical jargon: affine permutations) in which some \(p(j)\lt j\) exceptions are allowed. I didn't quite understand whether the decoration was needed, whether it affects something, why it was needed, and what the decoration affects. The paper may be needed but I will try to watch the talk again. Moreover, I had thought that the affine permutation is just an ordinary permutation of a very special type, \(p(j)=aj+b\) modulo \(n\) in \(\ZZ_n\), not a general permutation with some extra structure.

Nima discussed the special subset of "bipartite graphs". In those diagrams, only edges connecting one black vertex and one white vertex are allowed. Especially for those graphs, one may discuss various nice operations such as the "square move" – changing the colors around a "box" inside the diagram from black-white-black-white to white-black-white-black.

An annoying technical feature of these Munich videos is that one doesn't see the laser pointer on the PDF slides so we don't really know what object on the slide the speaker is talking about in a given sentence. It surely reduces the degree of understanding for those of us who watch the talks on the web.


Nima argues that the "change of consecutive vectors" is an important procedure and he slowly gets to the Grassmannian – the set of lower-dimensional (hyper)planes within a higher-dimensional (hyper)space. In particular, this whole story is about the positive Grassmannians. You may see that Alexander Postnikov, an amazing mathematician and a co-author in this permutation twistor business, is routinely talking about and teaching the positive Grassmannians.

To see why the positive Grassmannians are yet another equivalent way to describe the graphs, one must realize that there exists an explicit integral of delta-functions that guarantees that three spinors \(\lambda\) are proportional to each other – which follows from the momentum conservation for 3 vectors (well, either this follows, or the same statement for the mirror \(\tilde\lambda\) spinors follows from that).

This explicit integral enforcing the proportionality of the three spinors \(\lambda\) allows one to glue the things together and see the emergence of the Grassmannian. In the mathematicians' jargon, this gluing is known as the amalgamation. While you could have a priori thought that the "life" occurs at the vertices, these arguments lead you to realize that "life" takes places at the faces of the diagram – or at least the edges (modulo something).

Nima discussed how to add orientation to all the edges. White and black vertices must have 2-in-and-1-out or 2-out-and-1-in arrows, respectively. (Sorry if I reverted the rule again.) He finally offers some map from a diagram to a Grassmannian.

There are various simplifications. If you modify a diagram and replace it by equivalent ones, you may create a "bubble" – a propagator with a 1-loop "self-energy" with one black vertex and one white vertex in the loop. It's possible to get rid of such components of the diagrams.

Again, a key new insight is that a permutation of the external gluons is an invariant way to describe a particular diagram.

In this new perspective, the BCFW recursion rules may be viewed as a particular way to construct the Feynman-like on-shell diagram with black and white vertices out of a given permutation. There exist many other ways, too. I suppose that the total amplitude is given as a sum over all permutations. At this point, I have an obvious temptation to represent the whole structure in an analogous way as matrix string theory without a time dimension (action-based instead of Hamiltonian-based) but I will have to think about all these things in the wake of all the information that I learned from the talk and didn't know from my previous exposure to these cute ideas. ;-)

Now, the dual conformal invariance is getting manifest in this new permutation-or-positive-Grassmannian language. While the dual conformal invariance would rearrange the black-and-white diagrams in very complicated ways, it modifies the permutation in a very well-defined and simple way. Some jumps over two are involved but I haven't caught the exact answer so I will refrain from confusing you with possibly invalid caricatures of the right rule.

This positive Grassmannian construction allows one to write all the loop amplitudes in the form\[

\text{d log} (\dots)\,\text{d log} (\dots)\,\text{d log} (\dots)\,\text{d log} (\dots)\,

\] and this form may be obtained by some clever change of variables that (and whose existence) couldn't have been clear from the beginning and that was discovered by an accident (after some months of confusing interactions between physicists and mathematicians, they realized that they had been talking about the same thing – but the physicists had paradoxically had the more mathematically elegant, coordinate-independent form of the object). Nevertheless, this change of variables remarkably clarifies many hidden pattern in the amplitudes.

Nima said that Feynman's evaluation of the loop diagrams may be thought of as an evaluation using lots of auxiliary spaces to perform the integrals. In some sense, their twistor construction uses no auxiliary spaces at all: it reduces all the integrals to some low-dimensional loci. At the beginning, you could think that the d-log-like structure may simplify the integrals into "nothing" or "points". But when you do things properly, you will always find out that the integrals simplify but not to constants. They reduce to low-dimensional integrals such as polylogs.

For example, the d-log-to-the-fourth structure of the 4-point amplitude above is related to the geometric fact that there are 2 null-separated points in spacetime from the 4 points associated with the gluons we scatter.

At the end of the talk, Nima had to speed up a bit so many of his statements looked like intriguing demos. For example, we learned that the Yang-Baxter equations as well as the ABJM stuff (another, membrane minirevolution!) became special cases of this twistor business.

The folks were told that loops are needed reconstruct the world sheet theory out of the planar diagrams – and yes, as Nima confirmed in an answer to a question, everything he discussed was about the planar diagrams only.

The locality and unitarity are not the stars of the show. It would still be good to see why they're right – well, a proof of the equivalence to the normal ways to calculate the diagrams is probably enough for that. (Of course, by seeing the right properties that determine the amplitudes uniquely, they know that the final S-matrix is the same. It's still plausible that the equivalence may be shown at a "more localized" level, I think.)

Nima thinks that those structures should arise from the \((2,0)\) theory in six dimensions; it seems likely that this comment largely boils down to a wishful thinking at this point.

Questions and answers

The first physicist who asked – a Russian one I guess – asked the same question he asked on Friday so he got the same answer and Nima made it clear that this is exactly what happened. He asked about the complexification. Nima said that all these things have to be considered in the complexified space but if you want to evaluate the amplitudes for the LHC or anything tangible of this sort, you may always convolute the formulae with real wave packets. That's it: that's Nima's efficient and fast way to deal with questions that are not exactly new and that are not exactly good, either. ;-)

The second question, a better one, was about the restrictions on gauge groups, the number of colors, planarity, and similar things. So Nima said that only planar diagrams of the \(SU(N)\) gauge theory were considered at this point and repeated that this methodology is about localization of the integrals to low-dimensional loci in the spacetime (or related spaces).

He hopes to see a contact with the world sheet description. It's all very exciting but I may imagine that it will be ultimately seen to be a "purely" mathematical procedure dominated by the change of the variables on the world sheet (of the \(AdS_5\) dual string theory, used in the planar limit, which becomes "mostly topological" for these purposes) and a reorganization of the integrals and path integrals. Even if that's the case, it may still be a very important reorganization. They have already learned a great deal about interesting and "unified" geometric structures underlying amplitudes that are normally represented as sums of thousands of distinct Feynman diagrams.


  1. I'm still convinced that the weak isospin and weak hypercharge quantum numbers arise as the result of the permutation group. The various particles are different linear superpositions of the 6 permutations on three elements. That's a big space, C^6, but you reduce it by considering only the points that are idempotent, as is true of a pure density matrix representation of a state. There are a (more or less) finite number of results and the weak hypercharge / weak isospin values can be trivially calculated from the "n" and "k" values used (in the above lecture) to describe the permutations. That is in my paper, using (n,k) notation, you get to the variables in my section 5.2 by (0,0) = weak hypercharge and (1,2) = weak isospin. That is, the weak isospin of a particle with component b in the (0,0) direction is b.

    Too bad I'm quite happily working on digital electronics for LIGO.

  2. The toasted twistor makes me hungry (it looks really yummi) and I like the Micky Mouse explanations :-).

    But since I feel not so comfortable about the whole twistor business and these Grassmanians are definitively over my head :-/ ... I better rewatch Nima' s earlier twistor upraise instead of this one ... :-)

  3. Ive been growing a pretty decent-sized folder of many reviews and papers to understand this twistor minirevolution. Its definitely a really exciting set of clues, and Im personally pretty convinced of their relevance to understanding fundamental issues...

    The relationship between scattering amplitudes and Wilson loops in the dual spacetime was one thing that really floored me.

    I'll definitely be awaiting Nima's upcoming paper, which supposedly will deal more with the stringy aspects...

  4. Your temptation to represent the twistor structure in matrix string theory sounds interesting. In Witten's original twistor string paper (hep-th/0312171/) I recall Witten interpreting the gluon amplitudes in terms of (separated) genus zero curves of degree one (copies of CP^1~two-sphere). These curves in turn were interpreted as instantons in the stringy theory picture.

    In constructing connected amplitudes, it was assumed twistor fields (closed D-strings) were exchanged between the CP^1 instantons, giving the picture of two-spheres connected by collapsed cylinders. Such 2D instantons remind me of those in the U(N) SYM matrix string theory construction given back in '98 by Bonelli et al. (hep-th/9805071). Either way, your proposal looks promising.