Generalized scattering amplitudes, matroidal blade arrangements and the positive tropical Grassmannian

Seventy three years ago, Richard Feynman introduced a diagrammatic formalism in Quantum Field Theory (QFT) to organize the expansion of a scattering amplitude as a sum of elementary rational functions, labeled by graphs. These graphs, now called Feynman Diagrams, revolutionized theoretical physics and are still widely used today.

In using the scattering equations formalism to compute scattering amplitudes in terms of a Morse theoretic question posed on the Riemann sphere $\mathbb{CP}^{1}$, as introduced by Cachazo-He-Yuan in 2013, one can't help but feel that Feynman Diagrams should be living, breathing beings in their own right. Taking this idea seriously, in 2019, in joint work with Cachazo, Guevara and Mizera, we formulated an analog of the scattering equations on $\mathbb{CP}^{k-1}$, and introduced a generalized notion of scattering amplitudes: the generalized biadjoint scalar $m^{(k)}_n$. We found that, indeed, the set of Generalized Feynman Diagrams for the first new generalized amplitude $m^{(3)}_6$ should be understood as the set of maximal cones of the positive tropical Grassmannians $\text{Trop}^+G(3,6)$! This observation was quickly generalized to all $\text{Trop}^+G(k,n)$.

In this talk, we will explore how a novel structure in combinatorial geometry, the matroidal blade arrangement, can be used to study the rays of the positive tropical Grassmannian by selecting, essentially canonically, a very special basis of height functions over the hypersimplex $\Delta_{k,n}$, and, in particular, to organize and represent the poles appearing in Generalized Feynman Diagrams for $m^{(k)}_n$. Applications include an appearance of weak separation as a compatibility condition for certain positroidal subdivisions of $\Delta_{k,n}$.