BPS degeneracies and superconformal index in diverse dimensions

We present a unifying theme relating BPS partition functions and superconformal indices. In the case with complex supersymmetry central charges (as in N = 2 in d = 4 and 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 N = 1 in d = 5 (or the 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.