Positive Geometry, local triangulations and the dual of the Amplituhedron

by Cameron Langer (Penn State University)

Thursday 24 Jun 2021, 16:30 → 17:30 Europe/Berlin

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We initiate the systematic study of local positive spaces which arise in the context of the Amplituhedron for scattering amplitudes in planar maximally supersymmetric Yang-Mills theory. We show that all local positive spaces relevant for one-loop MHV amplitudes are characterized by certain sign-flip conditions and are associated with surprisingly simple logarithmic forms. In the maximal sign-flip case they are finite one-loop octagons. Particular combinations of sign-flip spaces can be glued into new local positive geometries. These correspond to local pentagon integrands that appear in the local expansion of the MHV one-loop amplitude. We show that, geometrically, these pentagons do not triangulate the original Amplituhedron space but rather its twin "Amplituhedron-Prime." This new geometry has the same boundary structure as the Amplituhedron (and therefore the same logarithmic form) but differs in the bulk as a geometric space. Interestingly, we find that the pentagons internally triangulate that dual space. This gives direct evidence that the chiral pentagons are natural building blocks for a yet-to-be discovered dual Amplituhedron.