Instantons from geometry in M2-brane sphere partition functions

Abstract

Can the geometry transverse to M2-branes capture effects that are exponentially small in their three-sphere ($S^3$) partition function? We investigate this question by extending beyond constant maps the proposal that M2-brane observables are encoded by equivariant topological strings on the transverse Calabi-Yau fourfold (CY$_4$). Our central object is the disk partition function of the CY$_4$ gauged linear sigma model, a quantum generalization of the equivariant volume. After a Gamma-function normalization and a Mellin transform, it reproduces the complete leading WKB grand potential in the examples admitting a suitable Fermi-gas description, including the R-charge and mass deformations compatible with that description. In particular, the transform recovers membrane-instanton series even for $\mathbb{C}^4$, which has no compact curves. For resolved CY$_4$ cones with compact curves, the match requires a quantum refinement of the classical mesonic constraint. We establish the exact equalities of the corresponding dressed disk partition functions with the classical spectral traces for ABJM, ADHM and a representative set of flavored M2-brane quiver theories. Away from the deformation loci where this match can be established, the same construction yields geometric predictions for the membrane-instanton sector that remain to be tested.