We define a new geometry obtained from the all-loop amplituhedron in $\mathcal{N}=4$ SYM by reducing its four-dimensional external and loop momenta to three dimensions. Focusing on the simplest four-point case, we provide 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, we present explicitly new results for the integrand up to five loops, which are much simpler than results in $\mathcal{N}=4$ 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. Our results suggest an unexpected relation between four-point amplitudes in these two theories.

• Received 28 April 2022
• Revised 30 September 2022
• Accepted 24 October 2022

DOI:https://doi.org/10.1103/PhysRevLett.129.221604

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Published by the American Physical Society