The BPS spectrum generator in 2D-4D systems

We apply the techniques provided by the recent works Gaiotto, Moore and Neitzke, to derive the most general spectrum generating functions for coupled 2d-4d A(1) theories of class S, in presence of surface and line defects. As an application of the result, some well-known BPS spectra are reproduced. Our results apply to a large class of coupled 2d-4d systems, the corresponding spectrum generating functions can be easily derived from our general expressions.