Nima Arkani-Hamed, Yuntao Bai, Song He, as well as Gongwang Yan – apologies to the bright Chinese folks for having represented them yesteryear a Western Farsi name, no impairment was intended as well as the evil of such abbreviations is appreciated on my side, it was only for the sake of simplification – accept published a 77-page-long preprint that lots of people were for certain waiting for for months:
Scattering Forms as well as the Positive Geometry of Kinematics, Color as well as the WorldsheetIn a recent paper, Nima promised some interrogation that shows the relevance of the generalized amplituhedron for perturbative string theory as well as this is the paper!
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I create hope that as many people as possible genuinely elbow grease to read the newspaper but permit me say a few basic as well as sketchy words.
The most of import novel – for most physicists – buzzword is the "associahedron" which these physicists purpose to unify the "amplituhedron", some variety of a polytope inward some variety of a twistor infinite from the twistor minirevolution, amongst other auxiliary spaces, such as the moduli infinite that nosotros integrate over inward perturbative calculations of opened upwards string scattering amplitudes.
Off-topic, Edward Witten: See an interview amongst a colleague of Arkani-Hamed virtually the (2,0) theory, M-theory, what's real, as well as tennis, amid other things. ;-)
At the laid out of the twistor minirevolution, nosotros withal idea that the shine twistor infinite was the most of import mathematical construction to focus upon. But the advances inward the people's agreement of diverse recursive formulae for the equations etc. accept led them to await at diverse damage inward the scattering amplitudes. These damage were identified amongst some combinatorial choices, discrete portions of something bigger, as well as they ultimately became ways to dissever a polytope to pieces etc.
So at the end, Nima as well as collaborators ended upwards seeing a scattering aAmplitude inward a guess theory as some integral of a simple, locally holomorphic component over a polytope living inward a higher-dimensional infinite – whose coordinates are twistor-like variables describing the external particles. The polytope may endure defined yesteryear many inequalities – these inequalities define the planes inward which the faces alive as well as they're the argue people beak virtually the "positivity" structures. And faces intersect at edges, edges intersect inward vertices or lower-dimensional edges, as well as then on. Arkani-Hamed as well as collaborators accept described diverse parts of the polytope using permutations and/or other combinatorial objects.
So the focus has moved from the shine twistor-like variables to the combinatorial choices how to fellowship or grouping variables, as well as those combinatorial choices may endure connected amongst things similar faces, edges, as well as other features of a higher-dimensional polytope.
Your humble correspondent, Andy Neitzke, as well as Sergei Gukov could encounter this duality betwixt "a unmarried shine description" as well as "several damage given yesteryear combinatorial choices" as an analogy to our proof of equivalence of connected as well as disconnected twistor prescriptions. The connected formulae due to Witten may endure shown equivalent to some disconnected ones. The erstwhile is simpler because it's ane term. The latter is simpler because the many damage involve "more linear" algebraic structures. Well, I hollo upwards it's non only an analogy – our newspaper is in all likelihood ane of the examples of the polytope analyses inward the modern papers yesteryear Arkani-Hamed et al.
At whatever rate, the polytopes such as the amplituhedron alive inward some auxiliary spaces as well as y'all could accept asked – as well as I accept asked – what is the large bargain here? We know lots of other examples inward which scattering amplitudes are written inward damage of integrals over some auxiliary parameters. Schwinger parameters are the criterion instance inward quantum plain theory – as well as the moduli spaces of Riemann surfaces inward perturbative string theory generalize them.
Now, Arkani-Hamed et al. genuinely claim to "unify" these structures. Well, everyone tin hand the axe say – as well as I accept said – that those things are analogous. But they acquire further, about 77 pages farther than that. ;-) They purpose the associahedron to disruptively unify the stringy moduli spaces, amplituhedron, Israeli Hebron, as well as the Standard Oil Chevron. Or at to the lowest degree most of those.
What is the associahedron? For my taste, the issue of fancy special words is besides high hither but it's non then difficult to acquire something virtually the associahedron. An associahedron, also known as a Stasheff polytope (proposed about 1963), is a fun visualization of ways to insert parentheses about some of \(n\) letters. The pic at the summit shows how it industrial plant for \(n=5\) letters.
There are ways to insert the parentheses e.g. as inward \(((ab)c)(de)\). These ways are identified amongst faces of a polytope. They may endure course of written report of "gradually converted" to each other if the parenthesizations are sufficiently similar. So edges – the intersections of side yesteryear side faces – gibe to a removal of a twosome of parenthesis from the parentheses describing either confront amid the two. And they only agree. If it tin hand the axe endure done, the border looks similar nosotros are using the associative dominion once, \(a(bc)=(ab)c\), as well as rearrange the parentheses a piffling chip (by ane minimum step) when nosotros switch from ane confront to another, side yesteryear side one. Because nosotros accept defined the "associative rule" to switch from ane confront of the polyhedron to another, the polyhedron is known as an associahedron.
The shocking thing derived to a greater extent than than one-half a century later on is that the associahedron – some visualization of parenthesizations that linguists may apply to \(n\) letters – may endure identified amongst the amplituhedron relevant for the calculation of some scattering amplitudes inward physics. In particular, the physical theory where the simplest associahedron from Wikipedia appears is cypher else than a cubic, \(\phi^3\), theory amongst bi-adjoint fields, Nima et al. show.
Great, some the combinatorial choices that parameterize some damage inward the scattering amplitudes of a cubic theory may endure identified amongst parenthesizations as well as hence aspects of a polytope.
Arkani-Hamed et al. present that a similar concept applies to opened upwards string theory. Tree-level opened upwards string amplitudes arise from distributing opened upwards string vertices inward some fellowship about a circle – the boundary of a disk-shaped basis sheet. The permutations of the opened upwards strings affair upwards to overall cyclic permutations which are immaterial, as well as some left-right reflection. So the positions affair upwards to \(SL(2,\RR)\) but the latter has to endure reduced yesteryear the permutation grouping reduced yesteryear the dihedral group, or something similar that. ;-)
These numerous damage inward the open-string formula that gibe to orderings of the opened upwards strings are also identified as some faces of a polytope. Each term (ordering of the opened upwards strings etc.) inward the open-string formula could endure considered to endure separate but the indicate of these Arkani-Hamed-style papers is to sentiment them as faces of ane object, ane polytope.
On summit of that, they also demonstrate that "color is kinematics". The color indices seem to endure ho-hum but they present that some Jacobi-like identity applies as to colors as well as kinematic coefficients. So the Chan-Paton factors at the opened upwards string destination points must also endure analogous as the degrees of liberty that y'all acquire from the fields living on the strings – similar the total momenta. At some qualitative level, it was ever clear – as well as I accept repeatedly tried to "derive" some Chan-Paton factors from to a greater extent than dynamical ones. But they present that the kinship relating these 2 dissimilar labels of the external states is deeper because some rather commutation Jacobi-like identities seem to apply to both variables equally. Let me rewrite a seven-term identity they abide by at both places:\[
S_{12}+S_{23}+S_{13} = S_1 + S_2 + S_3 + S_4
\] OK, some visualization of the dissimilar damage inward a scattering aAmplitude is made genuinely useful as well as "real", presented as a polytope, as well as the theories are described yesteryear a map from ane associahedron (linked to the auxiliary space) to some other ane (linked to the kinematic space, i.e. the infinite of all the Mandelstam-like variables labeling the momenta of the external particles). They are ambitious plenty to extend these analyses to loop diagrams but I don't know yet whether they claim to sympathise the consummate rules of the game.
Also, I am afraid that these constructions are alone applicable – or useful – for opened upwards string amplitudes, non unopen string ones (after all, there's alone ane closed-string Feynman diagram at each order, no "numerous faces"), which agency that all the theories they are able to beak over inward this way are non-gravitational theories. Influenza A virus subtype H5N1 guess theory may endure holographically dual to a gravitational theory inward the AdS infinite but I hollo upwards that all the guess theory variables computed yesteryear these methods are on-shell, as well as that's non plenty for gravity. You demand the total off-shell data virtually the guess theory to access the mass gravity.
But fifty-fifty if ane could "only" pull all auxiliary spaces that are useful for non-gravitational theories' scattering amplitudes inward some unified way, amongst some omnipresent geometric structures as well as Jacobi-like identities, it would endure fascinating. Does this formalism know anything virtually the stringy conformal symmetries at all? Or does it alone know virtually the topology of the multi-component moduli spaces of Riemann surfaces amongst punctures? This is extremely early on for me to say anything really deep because I accept alone spent a really pocket-sized amount of fourth dimension yesteryear studying the newspaper then far. I hope to sympathise it meliorate inward the futurity as well as I hope that something fifty-fifty deeper volition emerge out of these insights.