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.