Half-BPS surface operators of a 4d N=2 supersymmetric gauge theory can be constructed by coupling the 4d theory to a 2d N=(2,2) gauge theory on a surface. For 4d N=2 theories arising as the world-volume theory of M5 branes wrapping a Riemann surface, Jaume Gomis and I determined 2d N=(2,2) descriptions of arbitrary M2 branes ending on the M5 branes. This is tested thanks to the AGT correspondence which relates the sphere partition function of 4d N=2 theories and (two-dimensional) Toda CFT correlators: inserting a surface operator in the 4d N=2 theory amounts to inserting a so-called degenerate vertex operator on the Toda side. Gauge theory partition functions, known exactly using supersymmetric localization, yield new expressions for some Toda CFT conformal blocks. This leads to a Toda CFT braiding kernel which translates back to gauge theory as an S-duality domain wall for 4d N=2 SU(N) superconformal QCD.
