Relation between canonical forms and volume of polytopes

Question

Références:

Ref $1$: Henriette Elvang, Yu-tin Huang: Scattering Amplitudes

Ref $2$: Jaroslav Trnka: The Amplituhedron

[For simplicity, the notations of the $2$ refs have been merged]

The area of a triangle in $CP_2$, can be expressed, using dual space complex 3-dimensional quantities $Z_i$, as (Ref $1$, page $157$, formula $10.17$):

$$A = \frac{1}{2} \frac{\langle123\rangle^2}{\langle012\rangle\langle023\rangle\langle031\rangle}$$ where $i$ is for $Z_i$, $Z_0 = ^t(0,0,1)$, and $\langle abc \rangle=det(abc).$

On the other way, there is a "canonical form" (Ref $2$, page $26$):

$$\Omega_p= \frac{\langle Y dY dY\rangle \langle123\rangle^2}{\langle Y12\rangle\langle Y23\rangle\langle Y31\rangle},$$ where $Y$ represents a point in the interior of the triangle.

The relation between the $2$, (see Ref 2, pages $31,32$), seems to be an integration: $A =\int \delta(Y -Z_0) \Omega_p$

If the elements above are correct, that I don't understand is the utility of the canonical form, because in the integration, we keep only one point $Y = Z_0$, so the "integration" is somewhat "trivial", so why is used this presentation with the canonical form (which is linked to the grassmannian)?

Answer 1

There is a answer for a general polytope. For any polytope $P$, the canonical form associated with it is the volume of the dual polytope $\tilde{P}$: $$ \Omega^{(can)} [X, P] = \langle X d^nX\rangle \mathrm{Vol}[\tilde{P}] $$ Where $\langle X d^nX\rangle = \varepsilon_{I_1 I_2 \ldots I_{n+1}} X^{I_{1}} \ldots X^{I_{n}}$, $X$ - are homogeneous coordinates on projective space.