Generalized scattering amplitudes: planar kinematics, cyclic fixed points and two lattice polytopes

Clusters and Geometry
Event time: 
Friday, February 19, 2021 - 1:00pm
Nick Early
Speaker affiliation: 
Event description: 

The generalized biadjoint scalar amplitude, a highly structured rational function $m^{(k)}(\mathbb{I}_n,\mathbb{I}_n)$ on a dimension $\binom{n}{k}-n$ kinematic space calculated by summing over the critical points of a logarithmic potential function on the configuration space of $n$ generic points in $\mathbb{CP}^{k-1}$ modulo projective equivalence, was introduced recently by Cachazo-Early-Guevara-Mizera (CEGM); it grows quickly in size and complexity as $k$ increases. The poles of the amplitude can be as complicated as any coarsest regular positroidal subdivision of the hypersimplex $\Delta_{k,n}$, and any ray of the positive tropical Grassmannian; this presents quite an obstacle to any detailed general understanding of the amplitude!

In this talk, we shall discuss the evaluation of $m^{(k)}(\mathbb{I}_n,\mathbb{I}_n)$ at a particular Planar Kinematics (PK) point which has some surprising geometric and combinatorial properties. We find that the critical points of the PK potential function lift to equivalence classes of a certain subset of the critical points of the mirror superpotential as studied by Rietsch-Williams. In joint work with F. Cachazo, we conjectured that the $m^{(k)}(\mathbb{I}_n,\mathbb{I}_n)$ evaluates to the multi-dimensional Catalan number $C^{(k)}_{n-k}$ (e.g., the number of kx(n-k) standard Young tableaux); in the sequel, we used the scattering equations formalism to confirm our conjecture for (k,n) as large as (3,40) and (6,13). We finally discuss two interesting lattice polytopes which minimally bound the PK point.