Abstract
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 \( \mathfrak{s}\mathfrak{u}(2) \) super Yang-Mills coupled to the ℂℙ1 sigma model defect.
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Córdova, C., Gaiotto, D. & Shao, SH. Surface defect indices and 2d-4d BPS states. J. High Energ. Phys. 2017, 78 (2017). https://doi.org/10.1007/JHEP12(2017)078
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DOI: https://doi.org/10.1007/JHEP12(2017)078