  Alday, Gaiotto, and Tachikawa conjectured relations between certain 4d N=2
supersymmetric field theories and 2d Liouville conformal field theory. We study
generalizations of these relations to 4d theories with surface operators. For
one type of surface operators the corresponding 2d theory is the WZW model, and
for another type - the Liouville theory with insertions of extra degenerate
fields. We show that these two 4d theories with surface operators exhibit an IR
duality, which reflects the known relation (the so-called separation of
variables) between the conformal blocks of the WZW model and the Liouville
theory. Furthermore, we trace this IR duality to a brane creation construction
relating systems of M5 and M2 branes in M-theory. Finally, we show that this
duality may be expressed as an explicit relation between the generating
functions for the changes of variables between natural sets of Darboux
coordinates on the Hitchin moduli space.
