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A A question about the amplituhedron

  1. Nov 26, 2018 #1

    nrqed

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    Hi all,

    A quote from the paper "Into the amplituhedron"

    "The amplituhedron ##A_{n,k,L}## for n particle ##N^k MHV## amplitudes at L loops lives in G(k,k+4;L) which is the space of k- planes Y in k+4 dimensions... "


    I am trying to understand the meaning of the "4" here. Later they introduce the twistors Z and they say that there are n such twistors, each of dimension k+4. There is one twistor per particle, but why does each twistor have k+4 components? 4 I would understand (they probably have in mind spin 1/2 particles so there are four components for the helicity spinors), but why would the fact that we have the ##N^k## MHV increase the number of components of each particle's spinor?

    Thanks in advance.
     
  2. jcsd
  3. Nov 26, 2018 #2
    See page 14 of reference 1. The Z vectors are actually supertwistors, and the k extra components are Grassmann variables.
     
  4. Nov 26, 2018 #3

    nrqed

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    Ah!!Thank you!

    It is still strange to me that the number of these extra Grassmann variables are equal to k, which tells us how far we go beyond MHV. I guess I will have to follow the math to understand this.

    I hope you can also shed some light on a related (probably stupid) question: in their equation 2.2 (where the interior of a triangle is given as ##c_1 Z_1 + c_2 Z_2 + c_3 Z_3## with all c's positive definite), I don't see how that could cover the inside of the triangle, given that we are working in projective space.

    Let's say I have a triangle with vertices (0,0), (1,0) and (0,1). The point, say, (1/4,1/4) is inside, but this is the same point as (1/8,1/8) and so on. So I am missing something basic.

    Thanks again!
     
  5. Nov 27, 2018 #4
    See slide 14 of this talk by Trnka. Points in the 2d projective geometry correspond to rays in a 3d ordinary geometry. For the 3d perspective, Trnka attaches a dummy nonzero third coordinate to each vertex. If you then restrict yourself to positive combinations of those three-vectors, you are restricted to the 3d triangular 'cone' of rays passing through the triangle's interior.
     
  6. Nov 30, 2018 #5

    nrqed

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    Ah! That cleared things up, thank you so much.

    Sorry to keep asking questions but you have been so helpful (and I have nobody here to discuss with), let me also ask the following, in case the answer is short:

    A) In the case of a segment ##Y_1 Y_2##, they say that the matrix C must have positive minors. Is this simply to ensure that both ##Y_1## and ##Y_2## are inside the triangle formed by ##Z_1, Z_2, Z_3## (let's say when there are three particles)? I am probably not thinking about it the right way, though.

    B) What do they mean by "external data"? I thought it would be the Z's of the particles but that does not seem to be right.

    Thank you again for all your help!
     
  7. Dec 3, 2018 #6
    I think A) yes and B) it is those Zs, but I'm going to stop for a while because I don't know this material (though I want to), and don't have time to study it right now.
     
  8. Dec 3, 2018 #7

    nrqed

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    Thank you again very much for your help. It has been invaluable!

    Best,

    Patrick
     
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