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 …