  Recently, a physical derivation of the Alday-Gaiotto-Tachikawa correspondence
has been put forward. A crucial role is played by the complex Chern-Simons
theory arising in the 3d-3d correspondence, whose boundary modes lead to Toda
theory on a Riemann surface. We explore several features of this derivation and
subsequently argue that it can be extended to a generalization of the AGT
correspondence. The latter involves codimension two defects in six dimensions
that wrap the Riemann surface. We use a purely geometrical description of these
defects and find that the generalized AGT setup can be modeled in a pole region
using generalized conifolds. Furthermore, we argue that the ordinary conifold
clarifies several features of the derivation of the original AGT
correspondence.
