Positive Geometries for Scattering Amplitudes in N=4 SYM and ABJM

Abstract

This thesis investigates geometric descriptions of scattering amplitudes, with a specific focus on scattering amplitudes in N=4 SYM and ABJM theory. The recent development of the field of positive geometries provides us with a suitable framework for this endeavour. In particular, we will give a detailed account of the amplituhedron, the momentum amplituhedron, and the ABJM momentum amplituhedron. Alongside these geometries, we will also discuss the ABHY associahedron, which encapsulates tree-level scattering amplitudes in bi-adjoint scalar theory. We provide a detailed introduction to these positive geometries, which includes a comprehensive discussion of their structure. For the momentum amplituhedron, ABJM momentum amplituhedron, and ABHY associahedron we give a full stratification of their boundaries, which equivalently elucidates the singularity structure of the tree-level scattering amplitudes. Notably, we show that the ABJM momentum amplituhedron has an Euler characteristic equal to one. Furthermore, we explore the interconnections between these, and other, positive geometries. These connections are in part obtained via push forwards through the scattering equations. We develop techniques to calculate these push forwards which circumvents the necessity to solve the scattering equations explicitly. Beyond tree-level, we illustrate how positive geometries can be used to describe loop integrands in planar N=4 SYM and ABJM. A new framework is established to investigate these loop geometries in the space of dual momenta. The construction relies solely on lightcones and their intersections, and the framework simultaneously encompasses the loop level structure of the amplituhedron, momentum amplituhedron, and the ABJM momentum amplituhedron. This further leads to compact general formulae for all one-loop integrands in N=4 SYM and ABJM.