Non-critical M-Theory and the Resolved Conifold: Refinement and Nonperturbative Completion

We extend the Hořava–Keeler correspondence between finite-temperature non-critical M-theory and the resolved-conifold A-model to the refined theory. The grand potential of the stationary rotating non-critical M-theory vacuum reproduces the refined Gopakumar–Vafa expansion, with the angular chemical potential deforming the $\mathrm{\Omega }$-background away from the self-dual locus ${\text{eps}}_1=-{\text{eps}}_2$. At finite thermal radius, we establish an exact match with the nonperturbative conifold completions proposed by Hattab–Palti and by Chuang in the unrefined and refined cases, respectively. Both the one-particle Schwinger integrand and its integration cycle arise from the non-critical M-theory spectrum and resolvent, providing a microscopic spectral realization of these completions. The correspondence fixes the nonconstant BPS sector and the cubic local contribution, while the remaining polynomial ambiguity and the modulus-independent constant-map sector require separate normalization. We explain why the single primitive spinless multiplet of the resolved conifold makes this identification possible, and discuss what additional charge- and spin-dependent microscopic data would be needed for an extension to more general local Calabi–Yau geometries.