M2-brane partition functions and HD supergravity from equivariant volumes

We use the results of our companion paper arXiv:2502.20444, which explores the equivariant generalization of the constant-map contribution in topological string theory on toric Calabi--Yau manifolds $X$, to establish a holographic correspondence with M2-brane partition functions. We rigorously test this conjecture within the perturbative regime, incorporating all finite $N$ corrections on the field theory side. Our approach involves additional key details, such as incorporating effective 4d higher-derivative supergravity corrections to introduce refinement. A central result is the derivation of the Airy function representation for the squashed $S^3$ partition function of the field theory, for an arbitrary squashing parameter. We demonstrate that this Airy function structure is universal across all M2-brane models and provide a general expression in terms of the equivariant volume of $X$, incorporating the mesonic deformations corresponding to complexified masses. This expression is then evaluated explicitly for several examples, including ABJM theory, its flavored generalizations, circular quivers, and beyond, demonstrating agreement with the available field theory localization results. We extend the analysis to the superconformal and twisted indices of M2-brane models, and their spindle generalizations, leaving their full perturbative completion for future work. Finally, we explore avenues for generalizing these results to other brane systems, explicitly applying the idea to D3-branes.