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Hidden Permutation Symmetry of Squared Amplitudes in Aharony-Bergman-Jafferis-Maldacena Theory

Song He (何颂)^1,2,3,*, Canxin Shi (施灿欣)^1,†, Yichao Tang (唐一朝)^1,4,‡, and Yao-Qi Zhang (张耀奇)^1,4,§

¹Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
²School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, Hangzhou, Zhejiang 310024, China
³Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China
⁴School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

^*Contact author: songhe@itp.ac.cn
^†Contact author: shicanxin@itp.ac.cn
^‡Contact author: tangyichao@itp.ac.cn
^§Contact author: zhangyaoqi@itp.ac.cn

Phys. Rev. Lett. 135, 211601 – Published 17 November, 2025

DOI: https://doi.org/10.1103/zswh-bwc9

Abstract

We define the “squared amplitudes” in planar Aharony-Bergman-Jafferis-Maldacena theory, analogous to those in $N = 4$ super-Yang-Mills (SYM) theory. 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–Bogomol’nyi-Prasad-Sommerfield 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 that 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 satisfies, 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 Aharony-Bergman-Jafferis-Maldacena theory that unifies all squared amplitudes and satisfies physical constraints underlying these graphical rules.

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Phys. Rev. Lett. 135, 211601– Published 17 November, 2025

Vol. 135, Iss. 21 — 21 November 2025

Received 13 August 2025
Accepted 15 October 2025

DOI: https://doi.org/10.1103/zswh-bwc9

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

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Hidden Permutation Symmetry of Squared Amplitudes in Aharony-Bergman-Jafferis-Maldacena Theory

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