Positive geometry, local triangulations, and the dual of the Amplituhedron

Abstract

We initiate the systematic study of \emph{local positive spaces} which arise in the context of the Amplituhedron construction 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 \emph{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. On certain two-dimensional boundaries, where the Amplituhedron geometry reduces to a polygon, we check that both spaces map to the same dual polygon. Interestingly, we find that the pentagons internally triangulate that dual space. This gives a direct evidence that the chiral pentagons are natural building blocks for a yet-to-be discovered dual Amplituhedron.

117 pages, many figures

References in corpus (14)

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Cited by in corpus (14)

• Nonperturbative Negative Geometries: Amplitudes at Strong Coupling and the Amplituhedron
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes
• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• Multi-spin soft bootstrap and scalar-vector Galileon
• Amplituhedron-like geometries
• Internal boundaries of the loop amplituhedron
• Non-planar BCFW Grassmannian Geometries
• Kleiss-Kuijf Relations from Momentum Amplituhedron Geometry
• $\mathcal{N}=7$ On-Shell Diagrams and Supergravity Amplitudes in Momentum Twistor Space
• Maximal transcendental weight contribution of scattering amplitudes
• Poles At Infinity in On-shell Diagrams
• Positive Geometry for Stringy Scalar Amplitudes
• Extended DBI and its generalizations from graded soft theorems