RESUMEN
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.
RESUMEN
In this Letter, we study the equivalence between planar Ising networks and cells in the positive orthogonal Grassmannian. We present a microscopic construction based on amalgamation, which establishes the correspondence for any planar Ising network. The equivalence allows us to introduce two recursive methods for computing correlators of Ising networks. The first is based on duality moves, which generate networks belonging to the same cell in the Grassmannian. This leads to fractal lattices where the recursion formulas become the exact renormalization group equations of the effective couplings. The second, we use an amalgamation in which each iteration doubles the size of the seed lattice. This leads to an efficient way of computing the correlator where the complexity scales logarithmically with respect to the number of spin sites.