Aspects of Scattering Amplitudes and Moduli Space Localization

Understand

We elaborate on the recent proposal that intersection numbers of certain cohomology classes on the moduli space of genus-zero Riemann surfaces with punctures compute tree-level scattering amplitudes in quantum field theories.

• The relevant cohomology classes are twisted by representations of the fundamental group that describes how punctures braid around each other on the Riemann surface, which can be used to link it to the space of kinematic invariants.
• Intersection numbers of said cohomology classes, whose representatives we call twisted forms, can be shown to fully localize on the boundaries of the moduli space, which are in one-to-one map with Feynman diagrams.
• We prove that when twisted forms are logarithmic, their intersection numbers have a simple expansion in terms of trivalent Feynman diagrams allowing only for massless propagators on the internal and external lines.