Intersecting surface defects and two-dimensional CFT

We initiate the study of intersecting surface operators/defects in 4D quantum field theories (QFTs). We characterize these defects by coupled 4D/2D/0D theories constructed by coupling the degrees of freedom localized at a point and on intersecting surfaces in spacetime to each other and to the 4D QFT. We construct supersymmetric intersecting surface defects preserving just two supercharges in N=2 gauge theories. These defects are amenable to exact analysis by localization of the partition function of the underlying 4D/2D/0D QFT. We identify the 4D/2D/0D QFTs that describe intersecting surface operators in N=2 gauge theories realized by intersecting M2 branes ending on N M5 branes wrapping a Riemann surface. We conjecture and provide evidence for an explicit equivalence between the squashed four-sphere partition function of these intersecting defects and correlation functions in Liouville/Toda CFT with the insertion of arbitrary degenerate vertex operators, which are labeled by two representations of SU(N)