# Hexagon Functions Meet the Amplituhedron: Thinking Positive

I finished a new paper recently, it’s up on arXiv now.

This time, we’re collaborating with Jaroslav Trnka, of Amplituhedron fame, to investigate connections between the Amplituhedron and our hexagon function approach.

The Amplituhedron is a way to think about scattering amplitudes in our favorite toy model theory, N=4 super Yang-Mills. Specifically, it describes amplitudes as the “volume” of some geometric space.

Here’s something you might expect: if something is a volume, it should be positive, right? You can’t have a negative amount of space. So you’d naturally guess that these scattering amplitudes, if they’re really the “volume” of something, should be positive.

“Volume” is in quotation marks there for a reason, though, because the real story is a bit more complicated. The Amplituhedron isn’t literally the volume of some space, there are a bunch of other mathematical steps between the geometric story of the Amplituhedron on the one end and the final amplitude on the other. If it was literally a volume, calculating it would be quite a bit easier: mathematicians have gotten very talented at calculating volumes. But if it was literally a volume, it would have to be positive.

What our paper demonstrates is that, in the right regions (selected by the structure of the Amplituhedron), the amplitudes we’ve calculated so far are in fact positive. That first, basic requirement for the amplitude to actually literally be a volume is satisfied.

Of course, this doesn’t prove anything. There’s still a lot of work to do to actually find the thing the amplitude is the volume of, and this isn’t even proof that such a thing exists. It’s another, small piece of evidence. But it’s a reassuring one, and it’s nice to begin to link our approach with the Amplituhedron folks.

This week was the 75th birthday of John Schwarz, one of the founders of string theory and a discoverer of N=4 super Yang-Mills. We’ve dedicated the paper to him. His influence on the field, like the amplitudes of N=4 themselves, has been consistently positive.

## 14 thoughts on “Hexagon Functions Meet the Amplituhedron: Thinking Positive”

1. Thoglu

Cool, I ll try to understand something of that paper. This reminds of a small remark at the end of section 5.1 of https://arxiv.org/abs/1609.00008 (the “real-line” CHY paper), where the authors say that the resulting amplitudes might be interpreted as a volume of some sort… although it is not clear to me why .. Is there maybe a connection to this “real” CHY formalism in your paper/the Amplituhedron sector?

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Most popular accounts told the shorter story, that the amplitude is the volume of the amplituhedron. That isn’t quite what’s going on, but it isn’t just a conjecture either.

In technical terms, what Nima showed is that the amplitude is a form with logarithmic singularities on the boundary of the amplituhedron. This isn’t the same as being the volume: it also amounts to defining the entire amplitude with a geometrical object, but it isn’t as easy for mathematicians to calculate as a volume would be.

Beyond that, Nima thinks that there is some “dual amplituhedron” of which the amplitude actually is the volume. That part is conjecture, though he has been gathering evidence for it.

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2. ohwilleke

Off topic and a bit afield of your usual posting topics, but I’d love to hear your thoughts about the continuing non-discovery of smoking gun experimental evidence of glueballs.

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3. immediatism

Great work, Matt. The amplituhedron stuff and N=4 SYM generally are supremely fascinating, so its excellent to hear about progress seeing how all this fits together… Do keep us posted!

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4. Confused

I guess a more important question than is it volume would be: is it (i.e. the amplitude form) model- dependent? Volume form is obviously model-independent but is the reverse true? If both form and amplitu.. body end up to be model dependent the gain is less clear

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1. Confused

Amplitude as a general model independent form (as e.g. volume) on a model specific geometrical structure is one thing – it opens the path to geometrization of QFT. If both the form and the structure are model dependent, it’s less clear what is achieved as there’s no common framework that would apply to all (most, some) models.

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Ok, so to clarify, what you’re asking is if there were known amplithuedron-like geometrical objects for other QFTs, would that allow us to compute the amplitude immediately, or would there be some other step that would have to be modified to go from N=4 to more general cases?

“Form” here is a technical term, I’m not sure if that was clear. “A form with logarithmic singularities on the boundary” is a specific thing, there is one such form for each geometrical object. So whether or not the amplitude is a literal volume, there is a specific procedure for going from the Amplituhedron to the amplitude that doesn’t have any wiggle room.

Does that get closer to answering your question? If we had amplituhedra for other QFTs, we’d have a way of specifying any amplitude in those QFTs. We’re not particularly close to that, but regardless that’s an independent question unrelated to whether the amplitude is a volume.

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1. Confused

Yes, thanks. In that case I see less benefit in figuring out what sort of form it is for a specific theory as opposed to whether the same or similar consideration would apply to all or a class of QFT (as we have some hints e.g. from QCD). The geometrization approach thereby a complex algebraic problem is expressed in abstract algebraic/geometry terms has been successful with several difficult problems in math and if it could be attempted perhaps in a more complex/abstract way with QFT it could provide a hint to the next theory that could describe the reality. To sum, I don’t necessarily see how answering “this form is a volume” question would bring us closer to this goal (if that is the goal), shouldn’t we ask a more general one, what is the relation between this QFT and its geometric representation and how/can it be extended to other models?

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