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.

中文翻译：

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全环四点 Aharony-Bergman-Jafferis-Maldacena 振幅来自振幅面体的降维

我们定义了一个新的几何结构，该几何结构是从$\mathcal{否}=\mathrm{4个}$SYM 通过将其四维外部和循环动量减少到三个维度。着眼于最简单的四点情况，我们提供了强有力的证据表明这种“简化的振幅面体”的规范形式给出了 Aharony-Bergman-Jafferis-Maldacena 四点振幅的全环积函数。除了几何显示的各种全循环切割之外，我们还为最多五个循环的被积函数提供了明确的新结果，这比结果简单得多$\mathcal{否}=\mathrm{4个}$SYM。这种全循环简化的原因之一是只有很小一部分所谓的负几何在降维后幸存下来，这对应于二分图。我们的结果表明这两个理论中的四点振幅之间存在意想不到的关系。