  We study 4d $\mathcal{N}=2$ gauge theories with a co-dimension two full
surface operator, which exhibit a fascinating interplay of supersymmetric gauge
theories, equivariant Gromov-Witten theory and geometric representation theory.
For pure Yang-Mills and $\mathcal{N}=2^*$ theory, we describe a full surface
operator as the 4d gauge theory coupled to a 2d $\mathcal{N}=(2,2)$ gauge
theory. By supersymmetric localizations, we present the exact partition
functions of both 4d and 2d theories which satisfy integrable equations. In
addition, the form of the structure constants with a semi-degenerate field in
SL(N,R) WZNW model is predicted from one-loop determinants of 4d gauge theories
with a full surface operator via the AGT relation.
