A Hidden Permutation Symmetry of Squared Amplitudes in ABJM Theory

TL;DR

This paper uncovers a hidden permutation symmetry in squared amplitudes of ABJM theory, unifying them into a generating function with novel graphical properties and providing new explicit amplitude results.

Contribution

It introduces a generating function for squared amplitudes in ABJM theory with a hidden permutation symmetry, and develops graphical rules to bootstrap higher-point amplitudes.

Findings

Unifies squared amplitudes into a single generating function.

Identifies a hidden $S_N$ permutation symmetry in the dual space.

Provides explicit results for 10-point squared amplitudes.

Abstract

We define the square amplitudes in planar Aharony-Bergman-Jafferis-Maldacena theory (ABJM), analogous to that in $\mathcal{N}{=}4$ super-Yang-Mills theory (SYM). Surprisingly, the $n$-point $L$-loop integrands with fixed $N{:=}n{+}L$ are unified in a single generating function. Similar to the SYM four-point half-BPS correlator integrand, the generating function enjoys a hidden $S_N$ permutation symmetry in the dual space, allowing us to write it as a linear combination of weight-3 planar $f$-graphs. Remarkably, through Gram identities it can also be represented as a linear combination of bipartite $f$-graphs which manifest the important property that no odd-multiplicity amplitude exists in the theory. The generating function and these properties are explicitly checked against squared amplitudes for all $n$ with $N{=}4,6,8$. By drawing analogies with SYM, we conjecture some graphical rules the generating function satisfy, and exploit them to bootstrap a unique $N{=}10$ result, which provides new results for $n{=}10$ squared tree amplitudes, as well as integrands for $(n,L){=}(4,6),(6,4)$. Our results strongly suggest the existence of a "bipartite correlator" in ABJM theory that unifies all squared amplitudes and satisfies physical constraints underlying these graphical rules.