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

We define the “squared amplitudes” in planar Aharony-Bergman-Jafferis-Maldacena theory, analogous to those in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi mathvariant="script">N</a:mi> <a:mo>=</a:mo> <a:mn>4</a:mn> </a:mrow> </a:math> super-Yang-Mills (SYM) theory. Surprisingly, the <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mrow> <d:mi>n</d:mi> </d:mrow> </d:math> -point <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>L</f:mi> </f:math> -loop integrands with fixed <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mrow> <h:mi>N</h:mi> <h:mo>≔</h:mo> <h:mi>n</h:mi> <h:mo>+</h:mo> <h:mi>L</h:mi> </h:mrow> </h:math> 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 <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:msub> <j:mi>S</j:mi> <j:mi>N</j:mi> </j:msub> </j:math> permutation symmetry in the dual space, allowing us to write it as a linear combination of weight-3 planar <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mi>f</l:mi> </l:math> graphs. Remarkably, through Gram identities it can also be represented as a linear combination of “bipartite” <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:mi>f</n:mi> </n:math> 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 <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:mi>n</p:mi> </p:math> with <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:mi>N</r:mi> <r:mo>=</r:mo> <r:mn>4</r:mn> </r:math> , 6, 8. By drawing analogies with SYM, we conjecture some “graphical rules” the generating function satisfies, and exploit them to bootstrap a unique <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mi>N</t:mi> <t:mo>=</t:mo> <t:mn>10</t:mn> </t:math> result, which provides new results for <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:mi>n</v:mi> <v:mo>=</v:mo> <v:mn>10</v:mn> </v:math> squared tree amplitudes, as well as integrands for <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"> <x:mrow> <x:mo stretchy="false">(</x:mo> <x:mi>n</x:mi> <x:mo>,</x:mo> <x:mi>L</x:mi> <x:mo stretchy="false">)</x:mo> <x:mo>=</x:mo> <x:mo stretchy="false">(</x:mo> <x:mn>4</x:mn> <x:mo>,</x:mo> <x:mn>6</x:mn> <x:mo stretchy="false">)</x:mo> <x:mo>,</x:mo> <x:mo stretchy="false">(</x:mo> <x:mn>6</x:mn> <x:mo>,</x:mo> <x:mn>4</x:mn> <x:mo stretchy="false">)</x:mo> </x:mrow> </x:math> . 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.