We analytically study the Fermi-gas formulation of sphere correlation
functions of the Coulomb branch operators for 3d
<inline-formula><alternatives><tex-math>\mathcal{N}=4</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mstyle mathvariant="script"><mml:mi>𝒩</mml:mi></mml:mstyle><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>
ADHM theory with a gauge group <inline-formula><alternatives><tex-math>U(N)</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>U</mml:mi><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>,
an adjoint hypermultiplet and <inline-formula><alternatives><tex-math>l</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>l</mml:mi></mml:math></alternatives></inline-formula>
hypermultiplets which can describe a stack of
<inline-formula><alternatives><tex-math>N</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>N</mml:mi></mml:math></alternatives></inline-formula>
M2-branes at <inline-formula><alternatives><tex-math>A_{l-1}</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mi>l</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>
singularities. We find that the leading coefficients of the perturbative
grand canonical correlation functions are invariant under a hidden
triality symmetry conjectured from the twisted M-theory. The triality
symmetry also helps us to fix the next-to-leading corrections
analytically.