Toric Amplitudes and Universal Adjoints
Simon Telen
TL;DR
This work introduces toric amplitudes ${\rm Amp}_{\Sigma}$ and the universal adjoint ${\rm Adj}_{\Sigma}$ attached to a simplicial fan, defining adjoint hypersurfaces ${\cal A}_{\Sigma}$ in $\mathbb{P}^{n-1}$ that encode dual-volume and Warren adjoint structures. It develops a robust toolkit—covering product and restriction behavior, dual-volume interpretations, and explicit Fano-scheme and deformation analyses via the nef cone and irrelevant ideal—to study ${\cal A}_{\Sigma}$ in general and for key examples like quadrilaterals, pentagons, and the 3D ABHY associahedron. The paper provides concrete geometric descriptions of ${\cal A}_{\Sigma}$, its singular locus, and its linear spaces, including a unique interpolating adjoint for simple polytopes and irreducibility/singularity results for generic polygons; these results are supported by computational experiments. Altogether, the work links combinatorial algebraic geometry of toric data with physical amplitudes, offering a framework for exploring adjoint hypersurfaces, dual volumes, and Fano geometry in polyhedral settings.
Abstract
A toric amplitude is a rational function associated to a simplicial polyhedral fan. The definition is inspired by scattering amplitudes in particle physics. We prove algebraic properties of such amplitudes and study the geometry of their zero loci. These hypersurfaces play the role of Warren's adjoint via a dual volume interpretation. We investigate their Fano schemes and singular loci via the nef cone and toric irrelevant ideal of the fan.