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

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Summary

Introduction

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.

Amplituhedron geometry
Topological sign-flip definition of the Amplituhedron
One-loop MHV and MHV spaces
Boundaries of MHV amplitudes
Positivity and the dual Amplituhedron
Geometry of d log forms
From d log’s to geometry
No-go theorem for external triangulation
Sign-flip regions
Local geometries and the Amplituhedron-Prime
Chiral regions for boxes and pentagons
Two-dimensional projections
Gluing regions
Amplituhedron-Prime
Triangulation of the dual Amplituhedron
Dualizing polygons
Dual spaces of chiral pentagons
Conclusion
A Configuration of lines in momentum twistor space
B External triangulations
D Gluing local geometries from two-dimensional projections
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