$S^3$ partition functions and Equivariant CY${}_4$/CY${}_3$ correspondence from Quantum curves

Abstract

We study the perturbative large-$N$ expansion of the round three-sphere partition function in a class of M2-brane theories, including flavored SYM and ABJM theories as well as more general 3d theories admitting dual $(p,q)$ 5-brane web descriptions. Using the Fermi gas formalism and quantum curve techniques, we derive the Airy-function representation of the partition function and find exact agreement with predictions based on equivariant constant maps in topological string theory proposed in [1]. In particular, we provide affirmative tests of this proposal for the toric geometries $\mathbb{C}\times \mathcal{C}$ (the conifold), the cone over the Sasakian space $Q^{1,1,1}$, and $\mathbb{C}\times \mathrm{SPP}$ (the suspended pinch point). Motivated by a recent conjecture in [2], we further propose a novel equivariant correspondence between distinct toric Calabi-Yau manifolds of the form ${\mathrm{CY}}_4\leftrightarrow \mathbb{C}\times {\mathrm{CY}}_3$, arising from relations between the corresponding quantum curves under specific constraints. This correspondence suggests an equivariant extension and points toward a geometric origin of the topological string/spectral theory (TS/ST) correspondence, while offering new insight into the structure of the holographic duality.

73 pages, 11 figures