Abstract
We initiate the systematic study of 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 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.
Highlights
The Amplituhedron [1] is a geometric object encapsulating the tree-level amplitudes and allloop integrands of planar maximally supersymmetric Yang-Mills theory (N =4 sYM) [2, 3]
We show that all local positive spaces relevant for oneloop MHV amplitudes are characterized by certain sign-flip conditions and are associated with surprisingly simple logarithmic forms
Starting with the original d log form in eq (3.3) and checking the boundary structure of the 24 geometries arising from the respective sign choices for the entries of the d logs, we find that none of these spaces gives rise to a faithful geometry, i.e., these spaces always have certain additional geometric boundaries that do not appear as singularities of the form and are unacceptable to us
Summary
The Amplituhedron [1] is a geometric object encapsulating the tree-level amplitudes and allloop integrands of planar maximally supersymmetric Yang-Mills theory (N =4 sYM) [2, 3] It is a particular example of a positive geometry [4] and is defined as a certain geometric region in the space of positive external data. We show that the local expansion of the one-loop MHV amplitudes in terms of chiral pentagon integrals [72] can be naturally interpreted as the internal triangulation of the putative dual Amplituhedron We make this statement precise on two-dimensional boundaries of the full geometry, where the space reduces to polygons and the dualization procedure is well-defined.
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