We present a unifying theme relating BPS partition functions and superconformal indices. In the case with complex supersymmetry central charges (as in $\mathcal{N}=2$ in $d=4$ and $\mathcal{N}=(2,2)$ in $d=2$), the known results can be reinterpreted as the statement that the BPS partition functions can be used to compute a specialization of the superconformal indices. We argue that in the case with real central charge in the supersymmetry algebra, as in $\mathcal{N}=1$ in $d=5$ (or the $\mathcal{N}=2$ in $d=3$), the BPS degeneracy captures the full superconformal index. Furthermore, we argue that refined topological strings, which capture five-dimensional (5d) BPS degeneracies of M theory on Calabi-Yau 3-folds, can be used to compute a 5d supersymmetric index including in the sectors with three-dimensional defects for a large class of 5d superconformal theories. Moreover, we provide evidence that distinct Calabi–Yau singularities which are expected to lead to the same Superconformal field theory yield the same index.

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