Exact large $N$ expansion of $\mathcal{N}=4$ circular quiver Chern-Simons theories and squashing

In this work, we revisit the exact computation of the round sphere partition function of 3d $\mathcal{N}=4$ circular quiver Chern-Simons theories with mass and Fayet-Iliopoulos (FI) deformations. Utilizing the Fermi gas formalism, we derive the large $N$ expansion of the partition function and determine the Airy function structure, parameterized by three functions $C$, $B$ and $A$. We propose a novel closed-form expression for $A$ that incorporates the effects of FI parameters and satisfies various consistency constraints from quiver reductions. As an application, by using an accidental coincidence of the Fermi gas density matrices we extend our results to the squashed sphere partition function of $\mathcal{N}=4$ super Yang-Mills theories with an adjoint hypermultiplet and multiple fundamental hypermultiplets. Our findings provide further evidence for the universality of the Airy function structure in supersymmetric gauge theories of multiple M2-branes.