Aspects of Scattering Amplitudes and Moduli Space Localization

 It has been long known that intersection theory on the moduli space of punctured Riemann surfaces encodes observables in two-dimensional quantum gravity. It is natural to ask whether interacting theories could also admit a similar description. In the genus-zero case we put forward a twisted version of intersection theory on the moduli space and propose that it computes tree-level scattering amplitudes in a range of quantum field theories with a finite spectrum of particles. The resulting intersection numbers exhibit two alternative kinds of localization formulae. The first one receives contributions only from boundaries of the moduli space, thus leading to a Feynman diagram expansion, while the second one localizes on critical points of a certain Morse function.