M Theory Lesson 351

In the 1992 paper, Kapranov et al finish by detailing the example of three ternary quadrics. The full degree $12$ resultant is listed as $68$ bracket terms grouped according to $28$ different weights. The $14$ vertices of the associahedron appear as the singleton terms. In terms of brackets, these are

$-[145][246][356][456], [146][156][246][356], [145][245][256][356],$
$[145][246][346][345], [126]^2[156][356], [125]^2[256][356], [134]^2[246][346],$
$[136]^2[146][246], [145][245][235]^2, [145][345][234]^2,$
$[136]^2[126]^2, [125]^2[235]^2, [134]^2[234]^2, [123]^4$

where we observe how a zero area, like $[135]$, occurs as an unseen $1$ due to the zero volume exponent. These vertices thus depend on the metric structure of the triangle with vertices at $1$, $2$ and $3$. The so called toric specialisation takes bracket terms to monomials in $c_i$, the six coefficients in a quadratic form $f(X,Y,Z)$, via

$[i_1 i_2 \cdots i_k] \mapsto \textrm{det}(i_1, i_2, \cdots, i_k) c_1 c_2 \cdots c_k$

Here we see the volume factor go to an entropic coefficient, as in the term $[123]^4$ giving $256 c_{1}^{4}c_{2}^{4}c_{3}^{4}$. The sum of all $28$ such determinant terms turns out to be the product of principal minors for a $3 \times 3$ symmetric matrix with diagonal $2c_1, 2c_2, 2c_3$ and off diagonal terms $c_4$, $c_5$ and $c_6$. Thus a $3 \times 3$ matrix is associated to the six degree $2$ three qutrit functions.

Observe how the bracket monomials above could actually resemble associations of five objects, which usually label the associahedron by trees dual to the triangulated polygon. There are four nodes on a five leaved tree, and a total exponent of four on the bracket monomials. A tree node is labeled by a bracket, but a given bracket is allowed to repeat itself. For instance, if we imagine that $[123]$ labels the bottom node of the special tree for $[123]^4$, then $[123]$ stands for the fact that all subtrees above and to the left of the node are full binary trees. Then the mirror tree would be given by the term $-[145][246][356][456]$, and the spreading out of the $1,2,3$ reflects the fact that the left branch is now a minimal tree. Here we see metric structure being associated to tree shapes.