BPS/CFT correspondence IV: sigma models and defects in gauge theory

Quantum field theory $L_1$ on spacetime $X_1$ can be coupled to another quantum field theory $L_2$ on a spacetime $X_2$ via the third quantum field theory $L_{12}$ living on $X_{12}=X_1\cap X_2$. We explore several such constructions with two and four dimensional $X_1,X_2$'s and zero and two dimensional $X_{12}$'s, in the context of $\mathcal{N}=2$ supersymmetry, non-perturbative Dyson-Schwinger equations, and BPS/CFT correspondence. The companion paper will show that the BPZ and KZ equations of two dimensional conformal field theory are obeyed by the half-BPS surface defects in quiver $\mathcal{N}=2$ gauge theories.