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Internal boundaries of the loop amplituhedron
by Gabriele Dian, Paul Heslop, Alastair Stewart
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Gabriele Dian |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2207.12464v2 (pdf) |
| Date accepted: | 2023-07-24 |
| Date submitted: | 2022-11-17 16:03 |
| Submitted by: | Dian, Gabriele |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The strict definition of positive geometry implies that all maximal residues of its canonical form are $\pm 1$. We observe, however, that the loop integrand of the amplitude in planar $\mathcal{N}=4$ super Yang-Mills has maximal residues not equal to $\pm 1$. We find the reason for this is that deep in the boundary structure of the loop amplituhedron there are geometries which contain internal boundaries: codimension one defects separating two regions of opposite orientation. This phenomenon requires a generalisation of the concept of positive geometry and canonical form to include such internal boundaries and also suggests the utility of a further generalisation to `weighted positive geometries'. We re-examine the deepest cut of $\mathcal{N}=4$ amplitudes in light of this and obtain new all order residues.
Published as SciPost Phys. 15, 098 (2023)
Reports on this Submission
Report 1 by Jaroslav Trnka on 2023-1-11 (Invited Report)
- Cite as: Jaroslav Trnka, Report on arXiv:2207.12464v2, delivered 2023-01-11, doi: 10.21468/SciPost.Report.6513
Strengths
1 - Detailed study of the boundaries of the Amplituhedron including some all-loop order cases;
2 - Definition of weighted geometries, which is a new important concept to study positive geometries;
3 - Connection of these results to orientations and canonical forms.
Weaknesses
No apparent weakness.
Report
This is a very interesting paper with new original results which further advance our understanding of positive geometries and Amplituhedra. The paper is written very clearly with an emphasize on the main results, therefore I recommend the paper for a publication without any hesitations.
Requested changes
No changes requested.