The Geometry of BCFW for ABJM Loop Integrands

Abstract

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 $\text{Tr}({\varphi }^3)$ 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.

32 pages, 15 figures

References in corpus (18)

• The Amplituhedron
• Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet
• Positive Geometries and Canonical Forms
• Unwinding the Amplituhedron in Binary
• Tree-level Recursion Relation and Dual Superconformal Symmetry of the ABJM Theory
• The Momentum Amplituhedron
• ABJM amplitudes and the positive orthogonal grassmannian
• The Correlahedron
• Nonperturbative Negative Geometries: Amplitudes at Strong Coupling and the Amplituhedron
• Scalar-Scaffolded Gluons and the Combinatorial Origins of Yang-Mills Theory
• The orthogonal momentum amplituhedron and ABJM amplitudes
• The momentum amplituhedron of SYM and ABJM from twistor-string maps
• All-Loop Four-Point Aharony-Bergman-Jafferis-Maldacena Amplitudes from Dimensional Reduction of the Amplituhedron
• The Geometry of Cosmological Correlators
• The Loop Momentum Amplituhedron
• On the geometry of the orthogonal momentum amplituhedron
• Leading singularities and chambers of Correlahedron
• Positive Geometry for Stringy Scalar Amplitudes