  We conjecture a precise relationship between the Schur limit of the
superconformal index of four-dimensional $\mathcal{N}=2$ field theories, which
counts local operators, and the spectrum of BPS particles on the Coulomb
branch. We verify this conjecture for the special case of free field theories,
$\mathcal{N}=2$ QED, and $SU(2)$ gauge theory coupled to fundamental matter.
Assuming the validity of our proposal, we compute the Schur index of all
Argyres-Douglas theories. Our answers match expectations from the connection of
Schur operators with two-dimensional chiral algebras. Based on our results we
propose that the chiral algebra of the generalized Argyres-Douglas theory
$(A_{k-1},A_{N-1})$ with $k$ and $N$ coprime, is the vacuum sector of the
$(k,k+N)$ $W_{k}$ minimal model, and that the Schur index is the associated
vacuum character.
