Surface defect indices and 2d-4d BPS states

We conjecture a formula for the Schur index of four-dimensional N=2$$ \mathcal{N}=2 $$ 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 su2$$ \mathfrak{s}\mathfrak{u}(2) $$ super Yang-Mills coupled to the ℂℙ1 sigma model defect.