  We conjecture a formula for the Schur index of four-dimensional
$\mathcal{N}=2$ theories coupled to $(2,2)$ surface defects in terms of the
$2d$-$4d$ BPS spectrum in the Coulomb phase of the theory. The key ingredient
in our conjecture is a refined $2d$-$4d$ wall-crossing invariant, which we also
formulate. Our result intertwines recent conjectures expressing the
four-dimensional Schur index in terms of infrared BPS particles, with the
Cecotti-Vafa formula for limits of the elliptic genus in terms of
two-dimensional BPS solitons. We extend our discussion to framed $2d$-$4d$ BPS
states, and use this to demonstrate a general relationship between surface
defect indices and line defect indices. We illustrate our results in the
example of $SU(2)$ super Yang-Mills coupled to the $\mathbb{CP}^1$ sigma model
defect.
