The Geometry of BCFW for ABJM Loop Integrands

In this paper we investigate the loop-level geometry of ABJM theory from the perspective of lightcone geometries in dual space. This geometry admits a natural fibration, where one of the loop variables can be naturally interpreted as living in a fiber for each fixed point of a lower-loop geometry. When varying the latter, this leads us to the definition of $L$-loop half-chambers, defined such that `half' of the $(L+1)$-loop fiber remains unchanged. We provide a full classification of these half-chambers, and demonstrate a surprising bijection between $n$-point $L$-loop half-chambers and $L$-loop Feynman diagrams for a cubic scalar theory with $n/2$ particles. Consequently, the sum over $L$-loop half-chambers that computes the $n$-point ABJM amplitude is in direct correspondence with the sum over $L$-loop Feynman diagrams that computes the $(n/2)$-point amplitude of $\text{Tr}(\phi^3)$ theory. These Feynman diagrams are also realised geometrically in the structure of the loop fibers. Furthermore, we argue that the half-chamber expansion is equivalent to the loop-level BCFW recursion for ABJM, which arises naturally from our geometric construction. Finally, we will illustrate how $L$-loop chambers emerge as the intersection of two $L$-loop half-chambers, and we provide concrete examples of this construction.