  We use mirror symmetry, the refined holomorphic anomaly equation and
modularity properties of elliptic singularities to calculate the refined BPS
invariants of stable pairs on non-compact Calabi-Yau manifolds, based on del
Pezzo surfaces and elliptic surfaces, in particular the half K3. The BPS
numbers contribute naturally to the five-dimensional N=1 supersymmetric index
of M-theory, but they can be also interpreted in terms of the superconformal
index in six dimensions and upon dimensional reduction the generating functions
count N=2 Seiberg-Witten gauge theory instantons in four dimensions. Using the
M/F-theory uplift the additional information encoded in the spin content can be
used in an essential way to obtain information about BPS states in physical
systems associated to small instantons, tensionless strings, gauge symmetry
enhancement in F-theory by [p,q]-strings as well as M-strings.
