All-Loop Four-Point Aharony-Bergman-Jafferis-Maldacena Amplitudes from Dimensional Reduction of the Amplituhedron

We define a new geometry obtained from the all-loop amplituhedron in 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 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.