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

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.

Algebraic branch points at all loop orders from positive kinematics and wall crossing

“…Significant progress has been made in understanding the structure of N = 4 pSYM amplitudes beyond Feynman diagrams. At weak coupling, deep geometric structures, such as the amplituhedron, have emerged that provide both powerful computational techniques for computing integrands at any loop order and a radically different perspective on the nature of locality and unitarity [46][47][48][49][50]. At strong coupling, holographic calculations provide non-trivial predictions for the behavior of N = 4 pSYM amplitudes in the form of the BDS-ansatz [51][52][53] and its generalizations [30,54].…”

Kleiss-Kuijf relations from momentum amplituhedron geometry

“…The boundaries which are shared are oriented oppositely, and therefore they are not present in the oriented sum. Interestingly, this particular scenario has already been exploited in [10] for the amplituhedron-like geometries related to one-loop integrands. (II) Combinations of two geometries not giving a positive geometry.…”

$$ \mathcal{N} $$ = 7 On-shell diagrams and supergravity amplitudes in momentum twistor space

“…• Another interesting direction would be to extend the methods developed in this paper to loop amplitudes. When loop amplitudes of planar N = 4 SYM are represented in momentum twistor space, they can be expressed in terms of chiral pentagon in-JHEP01(2021)181 tegrals [39], which were recently proposed to be building blocks for a dual Amplituhedron [53]. It would be interesting to see if such integrals can be used to describe sugra amplitudes.…”