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We use mirror symmetry, the refined holomorphic anomaly equation andmodularity properties of elliptic singularities to calculate the refined BPSinvariants of stable pairs on non-compact Calabi-Yau manifolds, based on delPezzo surfaces and elliptic surfaces, in particular the half K3. The BPSnumbers contribute naturally to the five-dimensional N=1 supersymmetric indexof M-theory, but they can be also interpreted in terms of the superconformalindex in six dimensions and upon dimensional reduction the generating functionscount N=2 Seiberg-Witten gauge theory instantons in four dimensions. Using theM/F-theory uplift the additional information encoded in the spin content can beused in an essential way to obtain information about BPS states in physicalsystems associated to small instantons, tensionless strings, gauge symmetryenhancement in F-theory by [p,q]-strings as well as M-strings.

Refined stable pair invariants for E-, M- and [p,q]-strings