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Associahedron: Arkani-Hamed + 3 men unify auxiliary spaces for amplitudes

Linguistics, brackets, Jacobi's patents unify hedrons, Hebrons, Chevrons, amplituhedrons, colors, and open strings ;-)

Nima Arkani-Hamed, Yuntao Bai, Song He, and Gongwang Yan – apologies to the brilliant Chinese folks for having represented them by a Persian name, no harm was intended and the evil of such abbreviations is appreciated on my side, it was just for the sake of simplification – have published a 77-page-long preprint that lots of people were surely waiting for for months:

Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet
In a recent paper, Nima promised some research that shows the relevance of the generalized amplituhedron for perturbative string theory and this is the paper!

I do hope that as many people as possible actually try to read the paper but let me say a few basic and sketchy words.

The most important new – for most physicists – buzzword is the "associahedron" which these physicists use to unify the "amplituhedron", some kind of a polytope in some kind of a twistor space from the twistor minirevolution, with other auxiliary spaces, such as the moduli space that we integrate over in perturbative calculations of open string scattering amplitudes.

Off-topic, Edward Witten: See an interview with a colleague of Arkani-Hamed about the (2,0) theory, M-theory, what's real, and tennis, among other things. ;-)

At the beginning of the twistor minirevolution, we still thought that the smooth twistor space was the most important mathematical structure to focus upon. But the advances in the people's understanding of various recursive formulae for the equations etc. have led them to look at various terms in the scattering amplitudes. These terms were identified with some combinatorial choices, discrete portions of something bigger, and they ultimately became ways to divide a polytope to pieces etc.

So at the end, Nima and collaborators ended up seeing a scattering amplitude in a gauge theory as some integral of a simple, locally holomorphic function over a polytope living in a higher-dimensional space – whose coordinates are twistor-like variables describing the external particles. The polytope may be defined by many inequalities – these inequalities define the planes in which the faces live and they're the reason people talk about the "positivity" structures. And faces intersect at edges, edges intersect in vertices or lower-dimensional edges, and so on. Arkani-Hamed and collaborators have described various parts of the polytope using permutations and/or other combinatorial objects.

So the focus has moved from the smooth twistor-like variables to the combinatorial choices how to order or group variables, and those combinatorial choices may be connected with things like faces, edges, and other features of a higher-dimensional polytope.

Your humble correspondent, Andy Neitzke, and Sergei Gukov could see this duality between "a single smooth description" and "several terms given by combinatorial choices" as an analogy to our proof of equivalence of connected and disconnected twistor prescriptions. The connected formulae due to Witten may be shown equivalent to some disconnected ones. The former is simpler because it's one term. The latter is simpler because the many terms involve "more linear" algebraic structures. Well, I think it's not just an analogy – our paper is probably one of the examples of the polytope analyses in the modern papers by Arkani-Hamed et al.

At any rate, the polytopes such as the amplituhedron live in some auxiliary spaces and you could have asked – and I have asked – what is the big deal here? We know lots of other examples in which scattering amplitudes are written in terms of integrals over some auxiliary parameters. Schwinger parameters are the standard example in quantum field theory – and the moduli spaces of Riemann surfaces in perturbative string theory generalize them.

Now, Arkani-Hamed et al. really claim to "unify" these structures. Well, everyone can say – and I have said – that those things are analogous. But they get further, approximately 77 pages further than that. ;-) They use the associahedron to disruptively unify the stringy moduli spaces, amplituhedron, Israeli Hebron, and the Standard Oil Chevron. Or at least most of those.

What is the associahedron? For my taste, the number of fancy special words is too high here but it's not so hard to learn something about the associahedron. An associahedron, also known as a Stasheff polytope (proposed around 1963), is a fun visualization of ways to insert parentheses around some of \(n\) letters. The picture at the top shows how it works for \(n=5\) letters.

There are ways to insert the parentheses e.g. as in \(((ab)c)(de)\). These ways are identified with faces of a polytope. They may be sort of "gradually converted" to each other if the parenthesizations are sufficiently similar. So edges – the intersections of adjacent faces – correspond to a removal of a pair of parenthesis from the parentheses describing either face among the two. And they just agree. If it can be done, the edge looks like we are using the associative rule once, \(a(bc)=(ab)c\), and rearrange the parentheses a little bit (by one minimum step) when we switch from one face to another, adjacent one. Because we have defined the "associative rule" to switch from one face of the polyhedron to another, the polyhedron is known as an associahedron.

The shocking thing derived more than half a century later is that the associahedron – some visualization of parenthesizations that linguists may apply to \(n\) letters – may be identified with the amplituhedron relevant for the calculation of some scattering amplitudes in physics. In particular, the physical theory where the simplest associahedron from Wikipedia appears is nothing else than a cubic, \(\phi^3\), theory with bi-adjoint fields, Nima et al. show.

Great, some the combinatorial choices that parameterize some terms in the scattering amplitudes of a cubic theory may be identified with parenthesizations and therefore aspects of a polytope.

Arkani-Hamed et al. show that a similar concept applies to open string theory. Tree-level open string amplitudes arise from distributing open string vertices in some order around a circle – the boundary of a disk-shaped world sheet. The permutations of the open strings matter up to overall cyclic permutations which are immaterial, and some left-right reflection. So the positions matter up to \(SL(2,\RR)\) but the latter has to be reduced by the permutation group reduced by the dihedral group, or something like that. ;-)

These numerous terms in the open-string formula that correspond to orderings of the open strings are also identified as some faces of a polytope. Each term (ordering of the open strings etc.) in the open-string formula could be considered to be separate but the point of these Arkani-Hamed-style papers is to view them as faces of one object, one polytope.

On top of that, they also demonstrate that "color is kinematics". The color indices seem to be dull but they show that some Jacobi-like identity applies equally to colors and kinematic coefficients. So the Chan-Paton factors at the open string end points must also be analogous as the degrees of freedom that you get from the fields living on the strings – like the total momenta. At some qualitative level, it was always clear – and I have repeatedly tried to "derive" some Chan-Paton factors from more dynamical ones. But they show that the kinship relating these two different labels of the external states is deeper because some rather fundamental Jacobi-like identities seem to apply to both variables equally. Let me rewrite a seven-term identity they find at both places:\[

S_{12}+S_{23}+S_{13} = S_1 + S_2 + S_3 + S_4

\] OK, some visualization of the different terms in a scattering amplitude is made really useful and "real", presented as a polytope, and the theories are described by a map from one associahedron (linked to the auxiliary space) to another one (linked to the kinematic space, i.e. the space of all the Mandelstam-like variables labeling the momenta of the external particles). They are ambitious enough to extend these analyses to loop diagrams but I don't know yet whether they claim to understand the complete rules of the game.

Also, I am afraid that these constructions are only applicable – or useful – for open string amplitudes, not closed string ones (after all, there's only one closed-string Feynman diagram at each order, no "numerous faces"), which means that all the theories they are able to discuss in this way are non-gravitational theories. A gauge theory may be holographically dual to a gravitational theory in the AdS space but I think that all the gauge theory variables computed by these methods are on-shell, and that's not enough for gravity. You need the full off-shell information about the gauge theory to access the bulk gravity.

But even if one could "only" describe all auxiliary spaces that are useful for non-gravitational theories' scattering amplitudes in some unified way, with some omnipresent geometric structures and Jacobi-like identities, it would be fascinating. Does this formalism know anything about the stringy conformal symmetries at all? Or does it only know about the topology of the multi-component moduli spaces of Riemann surfaces with punctures? This is extremely early for me to say anything very deep because I have only spent a very small amount of time by studying the paper so far. I hope to understand it better in the future and I hope that something even deeper will emerge out of these insights.

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