Surface defect indices and 2d-4d BPS states

We conjecture a formula for the Schur index of four-dimensional N = 2 theories coupled to (2; 2) surface defects in terms of the 2d-4 d BPS spectrum in the Coulomb phase of the theory. The key ingredient in our conjecture is a re fi ned 2d-4 d wall-crossing invariant, which we also formulate. Our result intertwines recent conjectures expressing the fourdimensional 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 -4 d 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 CP1 sigma model defect.