Surface Defect Indices and 2d-4d BPS States

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.