A question about the amplituhedron

[nrqed]

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.

 

[mitchell porter]

See page 14 of reference 1. The Z vectors are actually supertwistors, and the k extra components are Grassmann variables.

 

[nrqed]

See page 14 of reference 1. The Z vectors are actually supertwistors, and the k extra components are Grassmann variables.

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!

 

[mitchell porter]

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.

 

[nrqed]

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.

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!

 

[mitchell porter]

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.

 

[nrqed]

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.

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