Positivity Sectors and the Amplituhedron

We initiate a detailed investigation into the assembly of simple amplituhedron-like building blocks to obtain spaces of physical interest. In particular, we describe the geometric process through which the building blocks, which we call positivity sectors, glue together to form the desired geometries. Positivity sectors are seen to naturally segment the space describing the Lth power of the one-loop amplitude. In this way, we obtain a good understanding of how the geometric complexity of the building blocks can be washed out in the formation of larger spaces. Conversely, the tools we develop allow us to form spaces of ever greater complexity, a process which is crucial to the construction of the amplituhedron from its triangulations, which remains an important open question. We present the full boundary structure of all positivity sectors related to the three-loop amplituhedron. We also construct a practical algorithm that achieves the desired geometric assembly of positivity sectors, and make available supporting Mathematica files containing the full boundary structure of all positivity sectors at three loops.