The amplituhedron in SYM and ABJM
In a recent letter (Phys. Rev. Lett. 129 (2022) 221604), S. He, Z. Li, Y. Zhang from ITP-CAS (Beijing) and C. Kuo from Taipei have discovered a new “amplituhedron” geometry; it is obtained from the all-loop amplituhedron in planar N=4 SYM by reducing its four-dimensional external and loop momenta to three dimensions. Focusing on the simplest four-point case, the authors have provided strong evidence that the canonical form of this “reduced amplituhedron” gives the all-loop integrand of the Aharony-Bergman-Jafferis-Maldacena four-point amplitude. In addition to various all-loop cuts manifested by the geometry, they have presented explicitly new results for the integrand up to five loops, which are much simpler than results in SYM. One of the reasons for such all-loop simplifications is that only a very small fraction of the so-called negative geometries survives the dimensional reduction, which corresponds to bipartite graphs. These results have suggested an unexpected relation between these two theories.