BCFW tilings and cluster adjacency for the amplituhedron

Journal Article · Fri Mar 21 00:00:00 EDT 2025 · Proceedings of the National Academy of Sciences of the United States of America

DOI:https://doi.org/10.1073/pnas.2408572122· OSTI ID:2575665

Even-Zohar, Chaim ^[1]; Lakrec, Tsviqa ^[2]; Parisi, Matteo ^[3]; Sherman-Bennett, Melissa ^[4]; Tessler, Ran ^[5]; Williams, Lauren ^[6]

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1. Technion, Haifa (Israel)
2. University of Geneva (Switzerland)
3. Harvard University, Cambridge, MA (United States); Institute for Advanced Study, Princeton, NJ (United States)
4. Massachusetts Institute of Technology, Cambridge, MA (United States)
5. Weizmann Institute of Science, Rehovot (Israel)
6. Harvard University, Cambridge, MA (United States)

In 2005, Britto, Cachazo, Feng, and Witten gave a recurrence (now known as the BCFW recurrence) for computing scattering amplitudes in N = 4 super Yang–Mills theory. Arkani-Hamed and Trnka subsequently introduced the amplituhedron to give a geometric interpretation of the BCFW recurrence. Arkani-Hamed and Trnka conjectured that each way of iterating the BCFW recurrence gives a “triangulation” or “tiling” of the m=4 amplituhedron. In this article, we prove the BCFW tiling conjecture of Arkani-Hamed and Trnka. We also prove the cluster adjacency conjecture for BCFW tiles of the amplituhedron, which says that facets of tiles are cut out by collections of compatible cluster variables for the Grassmannian Gr4,n. Moreover we show that each BCFW tile is the subset of the Grassmannian where certain cluster variables have particular signs.

• Research Organization: Institute for Advanced Study, Princeton, NJ (United States)
• Sponsoring Organization: Israel Science Foundation; National Science Foundation (NSF); Swiss National Science Foundation grant Dynamical Systems; USDOE
• Grant/Contract Number: SC0009988
• Other Award/Contract Number: 335/19; 335/19 and 1729/23; DMS-2152991; DMS-2103282; 188535; 1729/23; DMS-1854316
• OSTI ID: 2575665
• Journal Information: Proceedings of the National Academy of Sciences of the United States of America, Journal Name: Proceedings of the National Academy of Sciences of the United States of America Journal Issue: 12 Vol. 122; ISSN 1091-6490; ISSN 0027-8424
• Publisher: National Academy of Sciences
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
N = 4 super Yang–Mill
amplituhedron
cluster algebras
positive Grassmannian
scattering amplitudes