Positive geometry, local triangulations, and the dual of the Amplituhedron
[2];
- SLAC National Accelerator Lab., Menlo Park, CA (United States)
- Univ. of California, Davis, CA (United States). Center for Quantum Mathematics and Physics (QMAP); Pennsylvania State Univ., University Park, PA (United States). Institute for Gravitation and the Cosmos, Dept. of Physics
- Univ. of California, Davis, CA (United States). Center for Quantum Mathematics and Physics (QMAP)
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.
- Research Organization:
- SLAC National Accelerator Laboratory (SLAC), Menlo Park, CA (United States); Univ. of California, Davis, CA (United States)
- Sponsoring Organization:
- USDOE Office of Science (SC), Basic Energy Sciences (BES); USDOE Office of Science (SC), High Energy Physics (HEP)
- Grant/Contract Number:
- AC02-76SF00515; SC0009999
- OSTI ID:
- 1778936
- Alternate ID(s):
- OSTI ID: 1834291
- Journal Information:
- Journal of High Energy Physics (Online), Vol. 2021, Issue 1; ISSN 1029-8479
- Publisher:
- Springer BerlinCopyright Statement
- Country of Publication:
- United States
- Language:
- English
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