Preprint typeset in JHEP style- HYPER VERSION Wall-Crossing in Coupled 2d-4d Systems

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 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 su-persymmetric 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 S, that is, for those theories ob-tained 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 A1 theories of class S. Finally, we indi-cate how our results can be used to produce solutions to the A1 Hitchin equations on the Riemann surface C. ar X i