Refined BPS numbers on compact Calabi-Yau threefolds from Wilson loops

We relate the counting of refined BPS numbers on compact elliptically fibred Calabi-Yau threefolds X to Wilson loop expectations values in the gauge theories that emerge in various rigid local limits of the 5d supergravity theory defined by M-theory compactification on X. In these local limits X_∗ the volumes of curves in certain classes go to infinity, the corresponding very massive M2-brane states can be treated as Wilson loop particles and the refined topological string partition function on X becomes a sum of terms proportional to associated refined Wilson loop expectation values. The resulting ansatz for the complete refined topological partition function on X is written in terms of the proportionality coefficients which depend only on the ϵ deformations and the Wilson loop expectations values which satisfy holomorphic anomaly equations. Since the ansatz is quite restrictive and can be further constrained by the one-form symmetries and E-string type limits for large base curves, we can efficiently evaluate the refined BPS numbers on X, which we do explicitly for local gauge groups up to rank three and h₁₁(X) = 5. These refined BPS numbers pass an impressive number of consistency checks imposed by the direct counting of these numbers using the moduli space of one dimensional stable sheaves on X and give us numerical predictions for the complex structure dependency of the refined BPS numbers.