Subleading analysis for $S^3$ partition functions of $\mathcal{N}=2$ holographic SCFTs

We investigate the 3-sphere partition functions of various 3d $\mathcal{N}=2$ holographic SCFTs arising from the $N$ stack of M2-branes in the 't Hooft limit both analytically and numerically. We first employ a saddle point approximation to evaluate the free energy $F=-\log Z$ at the planar level, tracking the first subleading corrections in the large 't Hooft coupling $\lambda$ expansion. Subsequently, we improve these results by determining the planar free energy to all orders in the large $\lambda$ expansion via numerical analysis. Remarkably, the resulting planar free energies turn out to take a universal form, supporting a prediction that these $S^3$ partition functions are all given in terms of an Airy function even beyond the special cases where the Airy formulae were derived analytically in the literature; in this context we also present new Airy conjectures in several examples. The subleading behaviors we captured encode a part of quantum corrections to the M-theory path integrals around dual asymptotically Euclidean AdS$_4$ backgrounds with the corresponding internal manifolds through holographic duality.