Notes on Scattering Amplitudes as Differential Forms

Understand

Inspired by the idea of viewing amplitudes in ${\cal N}=4$ SYM as differential forms on momentum twistor space, we introduce differential forms on the space of spinor variables, which combine helicity amplitudes in any four-dimensional gauge theory as a single object.

• In this note we focus on such differential forms in ${\cal N}=4$ SYM, which can also be thought of as "bosonizing" superamplitudes in non-chiral superspace.
• Remarkably all tree-level amplitudes in ${\cal N}=4$ SYM combine to a $d\log$ form in spinor variables, which is given by pushforward of canonical forms of Grassmannian cells, the tree forms can also be obtained using BCFW or inverse-soft construction, and we present all-multiplicity expression for MHV and NMHV forms to illustrate their simplicity.
• Similarly all-loop planar integrands can be naturally written as $d\log$ forms in the Grassmannian/on-shell-diagram picture, and we expect the same to hold beyond the planar limit.