Abstract
We introduce a new wall-crossing formula which combines and generalizes the Cecotti-Vafa and Kontsevich-Soibelman formulas for supersymmetric 2d and 4d systems respectively. This 2d-4d wall-crossing formula governs the wall-crossing of BPS states in an \( \mathcal{N}=2 \) supersymmetric 4d gauge theory coupled to a supersymmetric surface defect. When the theory and defect are compactified on a circle, we get a 3d theory with a supersymmetric line operator, corresponding to a hyperholomorphic connection on a vector bundle over a hyperkähler space. The 2d-4d wall-crossing formula can be interpreted as a smoothness condition for this hyperholomorphic connection. We explain how the 2d-4d BPS spectrum can be determined for 4d theories of class \( \mathcal{S} \), that is, for those theories obtained by compactifying the six-dimensional (0, 2) theory with a partial topological twist on a punctured Riemann surface C. For such theories there are canonical surface defects. We illustrate with several examples in the case of A 1 theories of class \( \mathcal{S} \). Finally, we indicate how our results can be used to produce solutions to the A 1 Hitchin equations on the Riemann surface C.
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Gaiotto, D., Moore, G.W. & Neitzke, A. Wall-crossing in coupled 2d-4d systems. J. High Energ. Phys. 2012, 82 (2012). https://doi.org/10.1007/JHEP12(2012)082
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DOI: https://doi.org/10.1007/JHEP12(2012)082